-
Notifications
You must be signed in to change notification settings - Fork 89
/
mmhil.html
2896 lines (2587 loc) · 146 KB
/
mmhil.html
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN"
"https://www.w3.org/TR/html4/loose.dtd">
<HTML LANG="EN-US">
<HEAD>
<!-- improve mobile display -->
<META NAME="viewport" CONTENT="width=device-width, initial-scale=1.0">
<META HTTP-EQUIV="Content-Type"
CONTENT="text/html; charset=iso-8859-1">
<TITLE>Hilbert Space Explorer Home Page</TITLE>
<LINK REL="shortcut icon" HREF="favicon.ico" TYPE="image/x-icon">
<STYLE TYPE="text/css">
<!--
/* Math symbol image will be shifted down 4 pixels to align with normal
text for compatibility with various browsers. The old ALIGN=TOP for
math symbol images did not align in all browsers and should be deleted.
All other images must override this shift with STYLE="margin-bottom:0px".
(2-Oct-2015 nm) */
img { margin-bottom: -4px }
-->
</STYLE>
</HEAD>
<BODY BGCOLOR="#FFFFFF" STYLE="padding: 0px 8px">
<TABLE BORDER=0 CELLSPACING=0 CELLPADDING=0 WIDTH="100%">
<TR>
<TD ALIGN=LEFT VALIGN=TOP><A HREF="../index.html"><IMG SRC="atomic.gif"
BORDER=0
ALT="Metamath Home"
TITLE="Metamath Home"
HEIGHT=32 WIDTH=32 ALIGN=TOP STYLE="margin-bottom:0px"></A>
</TD>
<TD ALIGN=CENTER VALIGN=TOP><FONT SIZE="+3"
COLOR="#006633"><B>Hilbert Space Explorer Home Page</B></FONT>
</TD>
<TD NOWRAP ALIGN=RIGHT VALIGN=TOP><FONT SIZE=-2
FACE=sans-serif> <A HREF="chil.html">First ></A><BR><A
HREF="cdj3i.html">Last ></A></FONT>
</TD>
</TR>
<TR>
<TD COLSPAN=3 ALIGN=LEFT VALIGN=TOP><FONT SIZE=-2
FACE=sans-serif>
<A HREF="../mm.html">Mirrors</A> >
<A HREF="../index.html">Home</A> >
MPE Home >
<A HREF="mmtheorems.html">Th. List</A> >
<A HREF="mmrecent.html">Recent</A>
</FONT>
</TD>
</TR>
</TABLE>
<HR NOSHADE SIZE=1>
<B><FONT COLOR="#006633">Hilbert Space
Explorer</FONT></B>
<TABLE>
<TR><TD ROWSPAN=2>
<!--
The motivation for our study of quantum logic in the <A
HREF="">Quantum Logic Explorer</A> derives from the fact that the set
of closed subspaces of a Hilbert space, called <IMG SRC='_scrch.gif'
WIDTH=22 HEIGHT=19 ALT='CH'>, obeys the properties of an
orthomodular lattice.
-->
<I>Hilbert space</I> (<A
HREF="http://en.wikipedia.org/wiki/Hilbert_space">Wikipedia</A>
[external],
<!--
<A
HREF="http://planetmath.org/encyclopedia/HilbertSpace.html">PlanetMath</A>
[external],
-->
<A
HREF="http://mathworld.wolfram.com/HilbertSpace.html">MathWorld</A>
[external]) is a generalization of finite-dimensional
vector spaces to include vector spaces with infinite
dimensions. It provides a
foundation of quantum mechanics, and there is a strong physical and
philosophical motivation to study its properties. For example, the
properties of Hilbert space ultimately determine what kinds of
operations are theoretically possible in quantum computation.
</TD><TD><!-- <A HREF="#calcite"> --><IMG SRC='_calcite.jpg'
WIDTH=166 HEIGHT=113
ALT="Calcite" BORDER=0 STYLE="margin-bottom:0px"><!-- </A> --></TD>
</TR><TR><TD ALIGN=CENTER><FONT SIZE=-1><A
HREF="#calcite">Calcite</A></FONT></TD></TR>
</TABLE>
<HR NOSHADE SIZE=1>
<TABLE BORDER=0 WIDTH="100%"><TR><TD ROWSPAN=2>
<B><FONT COLOR="#006633">Contents of this page</FONT></B>
<MENU>
<LI> <A HREF="#follow">How to Follow the Proofs</A></LI>
<LI> <A HREF="#symbol">Symbol List</A>
<!--
<FONT SIZE=-2 FACE="Comic Sans MS" COLOR=ORANGE>Revised</FONT> <FONT
SIZE=-1><I>10-Sep-2009</I></FONT>
-->
</LI>
<LI> <A HREF="#approaches">Two Approaches to Hilbert Space</A>
</LI>
<LI> <A HREF="#axioms">The Axioms</A>
<FONT SIZE=-2 FACE="Comic Sans MS" COLOR=ORANGE>Revised</FONT> <FONT
SIZE=-1><I>10-Sep-2009</I></FONT>
</LI>
<LI> <A HREF="#definitions">Some Definitions</A></LI>
<LI> <A HREF="#theorems">Some Theorems</A></LI>
<!--
<LI> <A HREF="#choice">The Axiom of Choice</A></LI>
-->
<LI> <A HREF="#quantum">Quantum Logic</A>
<!--
<FONT SIZE=-2 FACE="Comic Sans MS" COLOR=ORANGE>Revised</FONT> <FONT
SIZE=-1><I>20-Feb-2006</I></FONT>
-->
</LI>
<LI> <A HREF="#next">What Next? (Orthoarguesian Law, etc.)</A></LI>
<!--
<LI> <A HREF="#state">States on Orthomodular Lattices</A></LI>
-->
<LI> <A HREF="#ref">References</A></LI>
</MENU>
</TD><TD VALIGN=TOP>
<B><FONT COLOR="#006633">Related pages</FONT></B>
<MENU>
<LI> <A HREF="mmtheorems.html">Table of Contents and Theorem List</A></LI>
<LI> <A HREF="mmbiblio.html">Bibliographic Cross-Reference</A></LI>
<LI><A HREF="mmset.html">Metamath Proof Explorer Home Page</A></LI>
<LI><A HREF="../qleuni/mmql.html">Quantum Logic Explorer Home Page</A></LI>
</MENU>
</TD></TR>
<TR><TD>
<!--
<P><CENTER><FONT COLOR="#006633"><I>The lyf so short,<BR>
the craft so long to lerne
</I><BR> —Chaucer</FONT></CENTER>
-->
</TD></TR>
</TABLE>
<HR NOSHADE SIZE=1><A NAME="follow"></A><B><FONT COLOR="#006633">How to
Follow the Proofs</FONT></B> We develop Hilbert space
theory as an extension of ZFC set theory, and many steps in various
proofs use results from set theory. To understand how to read the
proofs, see <A HREF="mmset.html#proofs">How Proofs Work</A> on the
Metamath Proof Explorer Home Page.
<HR NOSHADE SIZE=1><A NAME="symbol"></A><B><FONT COLOR="#006633">Symbol
List</FONT></B>
The chart below provides a quick reference for the new symbols
introduced in the Hilbert Space Explorer. The five symbols marked
"primitive" are postulated to have the properties specified by the <A
HREF="#axioms">axioms</A>, and the rest are defined in terms of them.
The complete list of all syntax elements, axioms, and definitions used
by the Hilbert Space Explorer pages, including those for the underlying
logic and ZFC set theory, is provided in the <A
HREF="mmdefinitions.html#startext">Definition List</A> (900K).
<P><CENTER><TABLE BORDER CELLSPACING=0 BGCOLOR="#EEFFFA">
<CAPTION><B>Symbol List for Hilbert Space</B></CAPTION>
<TR><TH>Symbol</TH>
<TH>Description</TH>
<TH>Link to Definition</TH></TR>
<TR><TD><IMG SRC='scrh.gif' WIDTH=19 HEIGHT=19 ALT='H~'></TD>
<TD>Hilbert space base set</TD>
<TD>(primitive)</TD></TR>
<TR><TD><IMG SRC='_pvh.gif' WIDTH=18 HEIGHT=19 ALT='+h'></TD>
<TD>vector addition</TD>
<TD>(primitive)</TD></TR>
<TR><TD><IMG SRC='_cds.gif' WIDTH=9 HEIGHT=19 ALT='.s'></TD>
<TD>scalar multiplication</TD>
<TD>(primitive)</TD></TR>
<TR><TD><IMG SRC='_0vh.gif' WIDTH=14 HEIGHT=19 ALT='0h'></TD>
<TD>zero vector</TD>
<TD>(primitive)</TD></TR>
<TR><TD><IMG SRC='_cdih.gif' WIDTH=13 HEIGHT=19 ALT='.ih'></TD>
<TD>inner (scalar) product</TD>
<TD>(primitive)</TD></TR>
<TR><TD><IMG SRC='_mvh.gif' WIDTH=16 HEIGHT=19 ALT='-h'></TD>
<TD>vector subtraction</TD>
<TD><A HREF="df-hvsub.html">df-hvsub</A></TD></TR>
<TR><TD><IMG SRC='_normh.gif' WIDTH=38 HEIGHT=19 ALT='normh'></TD>
<TD>norm of a vector</TD>
<TD><A HREF="df-hnorm.html">df-hnorm</A></TD></TR>
<TR><TD><IMG SRC='_cauchy.gif' WIDTH=47 HEIGHT=19 ALT='Cauchy'></TD>
<TD>set of Cauchy sequences</TD>
<TD><A HREF="df-hcau.html">df-hcau</A></TD></TR>
<TR><TD><IMG SRC='_squigv.gif' WIDTH=21 HEIGHT=19 ALT='~~>v'></TD>
<TD>convergence relation</TD>
<TD><A HREF="df-hlim.html">df-hlim</A></TD></TR>
<TR><TD><IMG SRC='_sh.gif' WIDTH=24 HEIGHT=19 ALT='SH'></TD>
<TD>set of subspaces</TD>
<TD><A HREF="df-sh.html">df-sh</A></TD></TR>
<TR><TD><IMG SRC='_scrch.gif' WIDTH=22 HEIGHT=19 ALT='CH'></TD>
<TD>set of closed subspaces</TD>
<TD><A HREF="df-ch.html">df-ch</A></TD></TR>
<TR><TD><IMG SRC='perp.gif' WIDTH=11 HEIGHT=19 ALT='_|_'></TD>
<TD>orthocomplement</TD>
<TD><A HREF="df-oc.html">df-oc</A></TD></TR>
<TR><TD><IMG SRC='_plh.gif' WIDTH=24 HEIGHT=19 ALT='+H'></TD>
<TD>subspace sum</TD>
<TD><A HREF="df-shs.html">df-shs</A></TD></TR>
<TR><TD><IMG SRC='_span.gif' WIDTH=31 HEIGHT=19 ALT='span'></TD>
<TD>subspace span</TD>
<TD><A HREF="df-span.html">df-span</A></TD></TR>
<TR><TD><IMG SRC='_veeh.gif' WIDTH=21 HEIGHT=19 ALT='vH'></TD>
<TD>join</TD>
<TD><A HREF="df-chj.html">df-chj</A></TD></TR>
<TR><TD><IMG SRC='_bigveeh.gif' WIDTH=23 HEIGHT=19 ALT='\/H'></TD>
<TD>supremum</TD>
<TD><A HREF="df-chsup.html">df-chsup</A></TD></TR>
<TR><TD><IMG SRC='_0h.gif' WIDTH=20 HEIGHT=19 ALT='0H'></TD>
<TD>zero subspace</TD>
<TD><A HREF="df-ch0.html">df-ch0</A></TD></TR>
<TR><TD><IMG SRC='_cch.gif' WIDTH=23 HEIGHT=19 ALT='C_H'></TD>
<TD>commutes relation</TD>
<TD><A HREF="df-cm.html">df-cm</A></TD></TR>
<TR><TD><IMG SRC='_plop.gif' WIDTH=25 HEIGHT=19 ALT='+op'></TD>
<TD>operator sum;<BR>definition of "operator"</TD>
<TD><A HREF="df-hosum.html">df-hosum</A></TD></TR>
<TR><TD><IMG SRC='_cdop.gif' WIDTH=16 HEIGHT=19 ALT='.op'></TD>
<TD>operator scalar product</TD>
<TD><A HREF="df-homul.html">df-homul</A></TD></TR>
<TR><TD><IMG SRC='_mop.gif' WIDTH=23 HEIGHT=19 ALT='-op'></TD>
<TD>operator difference</TD>
<TD><A HREF="df-hodif.html">df-hodif</A></TD></TR>
<TR><TD><IMG SRC='_plfn.gif' WIDTH=24 HEIGHT=19 ALT='+fn'></TD>
<TD>functional sum;<BR>definition of "functional"</TD>
<TD><A HREF="df-hfsum.html">df-hfsum</A></TD></TR>
<TR><TD><IMG SRC='_cdfn.gif' WIDTH=15 HEIGHT=19 ALT='.fn'></TD>
<TD>functional scalar product</TD>
<TD><A HREF="df-hfmul.html">df-hfmul</A></TD></TR>
<!--
<TR><TD><IMG SRC='_0op.gif' WIDTH=21 HEIGHT=19 ALT='0op'></TD>
-->
<TR><TD>0<SUB>hop</SUB> </TD>
<TD>zero operator</TD>
<TD><A HREF="df-h0op.html">df-h0op</A></TD></TR>
<TR><TD><IMG SRC='_iop.gif' WIDTH=18 HEIGHT=19 ALT='iop'></TD>
<TD>identity operator</TD>
<TD><A HREF="df-iop.html">df-iop</A></TD></TR>
<TR><TD>proj</TD>
<TD>projection operator (projector)</TD>
<TD><A HREF="df-pj.html">df-pj</A></TD></TR>
<TR><TD>norm<SUB>op</SUB></TD>
<TD>norm of an operator</TD>
<TD><A HREF="df-nmop.html">df-nmop</A></TD></TR>
<TR><TD>ConOp</TD>
<TD>set of continuous operators</TD>
<TD><A HREF="df-cnop.html">df-cnop</A></TD></TR>
<TR><TD>LinOp</TD>
<TD>set of linear operators</TD>
<TD><A HREF="df-lnop.html">df-lnop</A></TD></TR>
<TR><TD>BndLinOp</TD>
<TD>set of bounded linear operators</TD>
<TD><A HREF="df-bdop.html">df-bdop</A></TD></TR>
<TR><TD>UniOp</TD>
<TD>set of unitary operators</TD>
<TD><A HREF="df-unop.html">df-unop</A></TD></TR>
<TR><TD>HrmOp</TD>
<TD>set of Hermitian operators</TD>
<TD><A HREF="df-hmop.html">df-hmop</A></TD></TR>
<TR><TD>norm<SUB>fn</SUB></TD>
<TD>norm of a functional</TD>
<TD><A HREF="df-nmfn.html">df-nmfn</A></TD></TR>
<TR><TD>null</TD>
<TD>null space of a functional</TD>
<TD><A HREF="df-nlfn.html">df-nlfn</A></TD></TR>
<TR><TD>ConFn</TD>
<TD>set of continuous functionals</TD>
<TD><A HREF="df-cnfn.html">df-cnfn</A></TD></TR>
<TR><TD>LinFn</TD>
<TD>set of linear functionals</TD>
<TD><A HREF="df-lnfn.html">df-lnfn</A></TD></TR>
<TR><TD>adj<SUB>h</SUB></TD>
<TD>adjoint of an operator</TD>
<TD><A HREF="df-adjh.html">df-adjh</A></TD></TR>
<TR><TD>bra</TD>
<TD>Dirac "bra" of a vector</TD>
<TD><A HREF="df-bra.html">df-bra</A></TD></TR>
<TR><TD><IMG SRC='_leop.gif' WIDTH=24 HEIGHT=19 ALT='<_op'></TD>
<TD>ordering relation for positive operators</TD>
<TD><A HREF="df-leop.html">df-leop</A></TD></TR>
<TR><TD>ketbra</TD>
<TD>Dirac "ket-bra" (outer product) of two vectors</TD>
<TD><A HREF="df-kb.html">df-kb</A></TD></TR>
<TR><TD>eigvec</TD>
<TD>eigenvectors of an operator</TD>
<TD><A HREF="df-eigvec.html">df-eigvec</A></TD></TR>
<TR><TD>eigval</TD>
<TD>eigenvalue of an eigenvector</TD>
<TD><A HREF="df-eigval.html">df-eigval</A></TD></TR>
<TR><TD><IMG SRC='clambda.gif' WIDTH=11 HEIGHT=19 ALT='Lambda'></TD>
<TD>spectrum of an operator</TD>
<TD><A HREF="df-spec.html">df-spec</A></TD></TR>
<TR><TD><IMG SRC='_states.gif' WIDTH=40 HEIGHT=19 ALT='States'></TD>
<TD>set of states</TD>
<TD><A HREF="df-st.html">df-st</A></TD></TR>
<TR><TD>CHStates</TD>
<TD>set of (Mayet's) Hilbert-space-valued states</TD>
<TD><A HREF="df-hst.html">df-hst</A></TD></TR>
<TR><TD><IMG SRC='_atoms.gif' WIDTH=40 HEIGHT=19 ALT='Atoms'></TD>
<TD>set of atoms</TD>
<TD><A HREF="df-at.html">df-at</A></TD></TR>
<TR><TD><IMG SRC='lessdot.gif' WIDTH=11 HEIGHT=19 ALT='<o'></TD>
<TD>covering relation</TD>
<TD><A HREF="df-cv.html">df-cv</A></TD></TR>
<TR><TD><IMG SRC='_mh.gif' WIDTH=27 HEIGHT=19 ALT='MH'></TD>
<TD>modular pair relation</TD>
<TD><A HREF="df-md.html">df-md</A></TD></TR>
<TR><TD><IMG SRC='_mhast.gif' WIDTH=27 HEIGHT=19 ALT='MH*'></TD>
<TD>dual modular pair relation</TD>
<TD><A HREF="df-dmd.html">df-dmd</A></TD></TR>
</TABLE></CENTER>
<P>
<P><HR NOSHADE SIZE=1><A NAME="approaches"></A><B><FONT
COLOR="#006633">Two Approaches to Hilbert
Space</FONT></B> There are several ways to develop the
theory of Hilbert spaces. The purest way, philosophically, is to define
the class of all Hilbert spaces and use only the axioms of ZFC set
theory to derive its properties. That way we need to assume only the
axioms of ZFC (which in principle is all that is needed for essentially
all of mathematics, including the theory of Hilbert spaces). This is
done in the Metamath Proof Explorer with definition <A
HREF="df-hl.html">df-hl</A>.
<P>However, we chose separate axioms for the Hilbert Space Explorer for
several reasons. A practical problem with the pure ZFC approach is that
theorems becomes somewhat awkward to state and prove, since they usually
need additional hypotheses. Compare, for example, the ZFC-derived <A
HREF="hlcom.html">hlcom</A> with the Hilbert Space Explorer axiom <A
HREF="ax-hvcom.html">ax-hvcom</A>. Another advantage for a newcomer is that
the Hilbert Space Explorer states outright all of its axioms, so there
is nothing else to learn (aside from standard set theory tools to
manipulate them). In the Metamath Proof Explorer, on the other hand,
one needs to become familiar with the hierarchy of groups, topologies,
vector spaces, metric spaces, normed vector spaces, and Banach spaces
that leads to Hilbert spaces.
<P> If we want to use the Hilbert Space Explorer with any <I>fixed</I>
Hilbert space, such as the set of complex numbers (which, as it turns
out, is an example of a Hilbert space - see theorem <A
HREF="cnhl.html">cnhl</A>), a simple change to the axiomatization will
convert all theorems in the Hilbert Space Explorer to pure ZFC theorems.
A description of how this can be done is given in the comment for axiom
<A HREF="ax-hilex.html">ax-hilex</A>. On the other hand, if we want to prove
theorems involving relationships between Hilbert spaces, the Hilbert
Space Explorer may not be not suitable, but rewriting its proofs for the
general ZFC approach as needed is relatively straightforward.
(Actually, many such theorems can still be done in the Hilbert Space
Explorer itself using subspaces, each of which acts like a stand-alone
Hilbert Space.)
<P><HR NOSHADE SIZE=1><A NAME="axioms"></A><B><FONT COLOR="#006633">The
Axioms</FONT></B> In our separately axiomatized
approach of the Hilbert Space Explorer,
we postulate the existence of a new primitive fixed object, <IMG
SRC='scrh.gif' WIDTH=19 HEIGHT=19 ALT='H~'> (<A
HREF="chil.html">chil</A>), called the
Hilbert space base set, and add to ZFC set
theory explicit axioms for the properties of <IMG SRC='scrh.gif'
WIDTH=19 HEIGHT=19 ALT='H~'>.
The members of
<IMG SRC='scrh.gif' WIDTH=19 HEIGHT=19 ALT='H~'>
are called vectors, and they have the same
properties as the vectors you normally find in any linear algebra
textbook, except that the dimension (the number of basis vectors) is not
specified and may be infinite. In addition to
<IMG SRC='scrh.gif' WIDTH=19 HEIGHT=19 ALT='H~'>,
we postulate
the existence and properties of 4 more objects: a fixed zero vector
<IMG SRC='_0vh.gif' WIDTH=14 HEIGHT=19 ALT='0h'>
(<A HREF="c0v.html">c0v</A>); the operations of vector
addition
<IMG SRC='_pvh.gif' WIDTH=18 HEIGHT=19 ALT='+h'>
(<A HREF="cva.html">cva</A>) and scalar
multiplication
<IMG SRC='_cdh.gif' WIDTH=9 HEIGHT=19 ALT='.h'>
(<A HREF="csm.html">csm</A>); and finally, an inner product operation
<IMG SRC='_cdih.gif' WIDTH=13 HEIGHT=19 ALT='.ih'>
(<A HREF="csp.html">csp</A>). The five objects
<IMG SRC='scrh.gif' WIDTH=19 HEIGHT=19 ALT='H~'>,
<IMG SRC='_0vh.gif' WIDTH=14 HEIGHT=19 ALT='0h'>,
<IMG SRC='_pvh.gif' WIDTH=18 HEIGHT=19 ALT='+h'>,
<IMG SRC='_cdh.gif' WIDTH=9 HEIGHT=19 ALT='.h'>,
and
<IMG SRC='_cdih.gif' WIDTH=13 HEIGHT=19 ALT='.ih'>
are the complete
set of objects needed to describe Hilbert space. We will encounter
other objects as well, but all of them are defined either in terms of these
five, or as specific sets of set theory. For example, the object
<IMG SRC='bbc.gif' WIDTH=12 HEIGHT=19 ALT='CC'>
(the set of complex numbers <A HREF="cc.html">cc</A>) is
defined as a specific set of set theory.
<P>The page for each axiom below is accompanied by a precise breakdown
of its syntax. You can break the
syntax down into as much detail as you want by using the hyperlinks
in the syntax breakdown chart. Note our use of the
notation "<IMG SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('
ALIGN=TOP><IMG SRC='_ca.gif' WIDTH=11 HEIGHT=19 ALT='A'>
<IMG SRC='_cdih.gif' WIDTH=13 HEIGHT=19 ALT='.ih'>
<IMG SRC='_cb.gif'
WIDTH=12 HEIGHT=19 ALT='B'><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19
ALT=')'>" instead of the more common notation "<IMG
SRC='langle.gif' WIDTH=4 HEIGHT=19 ALT='<.' ><IMG
SRC='_ca.gif' WIDTH=11 HEIGHT=19 ALT='A'><IMG SRC='comma.gif'
WIDTH=4 HEIGHT=19 ALT=','><IMG SRC='_cb.gif' WIDTH=12
HEIGHT=19 ALT='B'><IMG SRC='rangle.gif' WIDTH=4 HEIGHT=19
ALT='>.' >" for inner products; the latter would
conflict with our notation for ordered pairs <A HREF="cop.html">cop</A>.
<P><CENTER><TABLE BORDER CELLSPACING=0 BGCOLOR="#EEFFFA">
<CAPTION><B>Axioms for Hilbert Space</B></CAPTION>
<TR ALIGN=LEFT><TD><A HREF="ax-hilex.html">ax-hilex</A></TD><TD>
<IMG SRC='_vdash.gif' WIDTH=10 HEIGHT=19 ALT='|-'> <IMG
SRC='scrh.gif' WIDTH=19 HEIGHT=19 ALT='H~'> <IMG SRC='in.gif'
WIDTH=10 HEIGHT=19 ALT='e.'> <IMG SRC='cv.gif' WIDTH=12 HEIGHT=19
ALT='V'></TD></TR>
<TR ALIGN=LEFT><TD><A HREF="ax-hfvadd.html">ax-hfvadd</A></TD><TD>
<IMG SRC='_vdash.gif' WIDTH=10 HEIGHT=19 ALT='|-'> <IMG
SRC='_pvh.gif' WIDTH=18 HEIGHT=19 ALT='+h'> <IMG SRC='colon.gif'
WIDTH=4 HEIGHT=19 ALT=':'><IMG SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('
ALIGN=TOP><IMG SRC='scrh.gif' WIDTH=19 HEIGHT=19 ALT='H~'> <IMG
SRC='times.gif' WIDTH=9 HEIGHT=19 ALT='X.'> <IMG SRC='scrh.gif'
WIDTH=19 HEIGHT=19 ALT='H~'><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19
ALT=')'><IMG SRC='longrightarrow.gif' WIDTH=23 HEIGHT=19 ALT='-->'
ALIGN=TOP><IMG SRC='scrh.gif' WIDTH=19 HEIGHT=19 ALT='H~'></TD></TR>
<TR ALIGN=LEFT><TD><A HREF="ax-hvcom.html">ax-hvcom</A></TD><TD>
<IMG SRC='_vdash.gif' WIDTH=10 HEIGHT=19 ALT='|-'> <IMG SRC='lp.gif'
WIDTH=5 HEIGHT=19 ALT='('><IMG SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('
ALIGN=TOP><IMG SRC='_ca.gif' WIDTH=11 HEIGHT=19 ALT='A'> <IMG
SRC='in.gif' WIDTH=10 HEIGHT=19 ALT='e.'> <IMG SRC='scrh.gif'
WIDTH=19 HEIGHT=19 ALT='H~'> <IMG SRC='wedge.gif' WIDTH=11 HEIGHT=19
ALT='/\'> <IMG SRC='_cb.gif' WIDTH=12 HEIGHT=19 ALT='B'>
<IMG SRC='in.gif' WIDTH=10 HEIGHT=19 ALT='e.'> <IMG SRC='scrh.gif'
WIDTH=19 HEIGHT=19 ALT='H~'><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19
ALT=')'> <IMG SRC='to.gif' WIDTH=15 HEIGHT=19 ALT='->'>
<IMG SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('><IMG SRC='_ca.gif'
WIDTH=11 HEIGHT=19 ALT='A'> <IMG SRC='_pvh.gif' WIDTH=18 HEIGHT=19
ALT='+h'> <IMG SRC='_cb.gif' WIDTH=12 HEIGHT=19 ALT='B'
ALIGN=TOP><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19 ALT=')'> <IMG
SRC='eq.gif' WIDTH=12 HEIGHT=19 ALT='='> <IMG SRC='lp.gif' WIDTH=5
HEIGHT=19 ALT='('><IMG SRC='_cb.gif' WIDTH=12 HEIGHT=19 ALT='B'
ALIGN=TOP> <IMG SRC='_pvh.gif' WIDTH=18 HEIGHT=19 ALT='+h'> <IMG
SRC='_ca.gif' WIDTH=11 HEIGHT=19 ALT='A'><IMG SRC='rp.gif' WIDTH=5
HEIGHT=19 ALT=')'><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19 ALT=')'
ALIGN=TOP></TD></TR>
<TR ALIGN=LEFT><TD><A HREF="ax-hvass.html">ax-hvass</A></TD><TD>
<IMG SRC='_vdash.gif' WIDTH=10 HEIGHT=19 ALT='|-'> <IMG SRC='lp.gif'
WIDTH=5 HEIGHT=19 ALT='('><IMG SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('
ALIGN=TOP><IMG SRC='_ca.gif' WIDTH=11 HEIGHT=19 ALT='A'> <IMG
SRC='in.gif' WIDTH=10 HEIGHT=19 ALT='e.'> <IMG SRC='scrh.gif'
WIDTH=19 HEIGHT=19 ALT='H~'> <IMG SRC='wedge.gif' WIDTH=11 HEIGHT=19
ALT='/\'> <IMG SRC='_cb.gif' WIDTH=12 HEIGHT=19 ALT='B'>
<IMG SRC='in.gif' WIDTH=10 HEIGHT=19 ALT='e.'> <IMG SRC='scrh.gif'
WIDTH=19 HEIGHT=19 ALT='H~'> <IMG SRC='wedge.gif' WIDTH=11 HEIGHT=19
ALT='/\'> <IMG SRC='_cc.gif' WIDTH=12 HEIGHT=19 ALT='C'>
<IMG SRC='in.gif' WIDTH=10 HEIGHT=19 ALT='e.'> <IMG SRC='scrh.gif'
WIDTH=19 HEIGHT=19 ALT='H~'><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19
ALT=')'> <IMG SRC='to.gif' WIDTH=15 HEIGHT=19 ALT='->'>
<IMG SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('><IMG SRC='lp.gif' WIDTH=5
HEIGHT=19 ALT='('><IMG SRC='_ca.gif' WIDTH=11 HEIGHT=19 ALT='A'
ALIGN=TOP> <IMG SRC='_pvh.gif' WIDTH=18 HEIGHT=19 ALT='+h'> <IMG
SRC='_cb.gif' WIDTH=12 HEIGHT=19 ALT='B'><IMG SRC='rp.gif' WIDTH=5
HEIGHT=19 ALT=')'> <IMG SRC='_pvh.gif' WIDTH=18 HEIGHT=19 ALT='+h'
ALIGN=TOP> <IMG SRC='_cc.gif' WIDTH=12 HEIGHT=19 ALT='C'><IMG
SRC='rp.gif' WIDTH=5 HEIGHT=19 ALT=')'> <IMG SRC='eq.gif' WIDTH=12
HEIGHT=19 ALT='='> <IMG SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('
ALIGN=TOP><IMG SRC='_ca.gif' WIDTH=11 HEIGHT=19 ALT='A'> <IMG
SRC='_pvh.gif' WIDTH=18 HEIGHT=19 ALT='+h'> <IMG SRC='lp.gif' WIDTH=5
HEIGHT=19 ALT='('><IMG SRC='_cb.gif' WIDTH=12 HEIGHT=19 ALT='B'
ALIGN=TOP> <IMG SRC='_pvh.gif' WIDTH=18 HEIGHT=19 ALT='+h'> <IMG
SRC='_cc.gif' WIDTH=12 HEIGHT=19 ALT='C'><IMG SRC='rp.gif' WIDTH=5
HEIGHT=19 ALT=')'><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19 ALT=')'
ALIGN=TOP><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19 ALT=')'></TD></TR>
<TR ALIGN=LEFT><TD><A HREF="ax-hv0cl.html">ax-hv0cl</A></TD><TD>
<IMG SRC='_vdash.gif' WIDTH=10 HEIGHT=19 ALT='|-'> <IMG
SRC='_0vh.gif' WIDTH=14 HEIGHT=19 ALT='0h'> <IMG SRC='in.gif'
WIDTH=10 HEIGHT=19
ALT='e.'> <IMG SRC='scrh.gif' WIDTH=19 HEIGHT=19 ALT='H~'
ALIGN=TOP></TD></TR>
<TR ALIGN=LEFT><TD><A HREF="ax-hvaddid.html">ax-hvaddid</A></TD><TD>
<IMG SRC='_vdash.gif' WIDTH=10 HEIGHT=19 ALT='|-'> <IMG SRC='lp.gif'
WIDTH=5 HEIGHT=19 ALT='('><IMG SRC='_ca.gif' WIDTH=11 HEIGHT=19
ALT='A'> <IMG SRC='in.gif' WIDTH=10 HEIGHT=19 ALT='e.'>
<IMG SRC='scrh.gif' WIDTH=19 HEIGHT=19 ALT='H~'> <IMG SRC='to.gif'
WIDTH=15 HEIGHT=19 ALT='->'> <IMG SRC='lp.gif' WIDTH=5 HEIGHT=19
ALT='('><IMG SRC='_ca.gif' WIDTH=11 HEIGHT=19 ALT='A'> <IMG
SRC='_pvh.gif' WIDTH=18 HEIGHT=19 ALT='+h'> <IMG SRC='_0vh.gif'
WIDTH=14 HEIGHT=19 ALT='0h'><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19
ALT=')'> <IMG SRC='eq.gif' WIDTH=12 HEIGHT=19 ALT='='> <IMG
SRC='_ca.gif' WIDTH=11 HEIGHT=19 ALT='A'><IMG SRC='rp.gif' WIDTH=5
HEIGHT=19 ALT=')'></TD></TR>
<TR ALIGN=LEFT><TD><A HREF="ax-hfvmul.html">ax-hfvmul</A></TD><TD>
<IMG SRC='_vdash.gif' WIDTH=10 HEIGHT=19 ALT='|-'> <IMG
SRC='_cdh.gif' WIDTH=9 HEIGHT=19 ALT='.h'> <IMG SRC='colon.gif'
WIDTH=4 HEIGHT=19 ALT=':'><IMG SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('
ALIGN=TOP><IMG SRC='bbc.gif' WIDTH=12 HEIGHT=19 ALT='CC'> <IMG
SRC='times.gif' WIDTH=9 HEIGHT=19 ALT='X.'> <IMG SRC='scrh.gif'
WIDTH=19 HEIGHT=19 ALT='H~'><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19
ALT=')'><IMG SRC='longrightarrow.gif' WIDTH=23 HEIGHT=19 ALT='-->'
ALIGN=TOP><IMG SRC='scrh.gif' WIDTH=19 HEIGHT=19 ALT='H~'></TD></TR>
<TR ALIGN=LEFT><TD><A HREF="ax-hvmulid.html">ax-hvmulid</A></TD><TD>
<IMG SRC='_vdash.gif' WIDTH=10 HEIGHT=19 ALT='|-'> <IMG SRC='lp.gif'
WIDTH=5 HEIGHT=19 ALT='('><IMG SRC='_ca.gif' WIDTH=11 HEIGHT=19
ALT='A'> <IMG SRC='in.gif' WIDTH=10 HEIGHT=19 ALT='e.'>
<IMG SRC='scrh.gif' WIDTH=19 HEIGHT=19 ALT='H~'> <IMG SRC='to.gif'
WIDTH=15 HEIGHT=19 ALT='->'> <IMG SRC='lp.gif' WIDTH=5 HEIGHT=19
ALT='('><IMG SRC='1.gif' WIDTH=7 HEIGHT=19 ALT='1'> <IMG
SRC='_cdh.gif' WIDTH=9 HEIGHT=19 ALT='.h'> <IMG SRC='_ca.gif'
WIDTH=11 HEIGHT=19 ALT='A'><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19
ALT=')'> <IMG SRC='eq.gif' WIDTH=12 HEIGHT=19 ALT='='> <IMG
SRC='_ca.gif' WIDTH=11 HEIGHT=19 ALT='A'><IMG SRC='rp.gif' WIDTH=5
HEIGHT=19 ALT=')'></TD></TR>
<TR ALIGN=LEFT><TD><A HREF="ax-hvmulass.html">ax-hvmulass</A></TD><TD>
<IMG SRC='_vdash.gif' WIDTH=10 HEIGHT=19 ALT='|-'> <IMG SRC='lp.gif'
WIDTH=5 HEIGHT=19 ALT='('><IMG SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('
ALIGN=TOP><IMG SRC='_ca.gif' WIDTH=11 HEIGHT=19 ALT='A'> <IMG
SRC='in.gif' WIDTH=10 HEIGHT=19 ALT='e.'> <IMG SRC='bbc.gif' WIDTH=12
HEIGHT=19 ALT='CC'> <IMG SRC='wedge.gif' WIDTH=11 HEIGHT=19 ALT='/\'
ALIGN=TOP> <IMG SRC='_cb.gif' WIDTH=12 HEIGHT=19 ALT='B'> <IMG
SRC='in.gif' WIDTH=10 HEIGHT=19 ALT='e.'> <IMG SRC='bbc.gif' WIDTH=12
HEIGHT=19 ALT='CC'> <IMG SRC='wedge.gif' WIDTH=11 HEIGHT=19 ALT='/\'
ALIGN=TOP> <IMG SRC='_cc.gif' WIDTH=12 HEIGHT=19 ALT='C'> <IMG
SRC='in.gif' WIDTH=10 HEIGHT=19 ALT='e.'> <IMG SRC='scrh.gif'
WIDTH=19 HEIGHT=19 ALT='H~'><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19
ALT=')'> <IMG SRC='to.gif' WIDTH=15 HEIGHT=19 ALT='->'>
<IMG SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('><IMG SRC='lp.gif' WIDTH=5
HEIGHT=19 ALT='('><IMG SRC='_ca.gif' WIDTH=11 HEIGHT=19 ALT='A'
ALIGN=TOP> <IMG SRC='cdot.gif' WIDTH=4 HEIGHT=19 ALT='x.'> <IMG
SRC='_cb.gif' WIDTH=12 HEIGHT=19 ALT='B'><IMG SRC='rp.gif' WIDTH=5
HEIGHT=19 ALT=')'> <IMG SRC='_cdh.gif' WIDTH=9 HEIGHT=19 ALT='.h'
ALIGN=TOP> <IMG SRC='_cc.gif' WIDTH=12 HEIGHT=19 ALT='C'><IMG
SRC='rp.gif' WIDTH=5 HEIGHT=19 ALT=')'> <IMG SRC='eq.gif' WIDTH=12
HEIGHT=19 ALT='='> <IMG SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('
ALIGN=TOP><IMG SRC='_ca.gif' WIDTH=11 HEIGHT=19 ALT='A'> <IMG
SRC='_cdh.gif' WIDTH=9 HEIGHT=19 ALT='.h'> <IMG SRC='lp.gif' WIDTH=5
HEIGHT=19 ALT='('><IMG SRC='_cb.gif' WIDTH=12 HEIGHT=19 ALT='B'
ALIGN=TOP> <IMG SRC='_cdh.gif' WIDTH=9 HEIGHT=19 ALT='.h'> <IMG
SRC='_cc.gif' WIDTH=12 HEIGHT=19 ALT='C'><IMG SRC='rp.gif' WIDTH=5
HEIGHT=19 ALT=')'><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19 ALT=')'
ALIGN=TOP><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19 ALT=')'></TD></TR>
<TR ALIGN=LEFT><TD><A HREF="ax-hvdistr1.html">ax-hvdistr1</A></TD><TD>
<IMG SRC='_vdash.gif' WIDTH=10 HEIGHT=19 ALT='|-'> <IMG SRC='lp.gif'
WIDTH=5 HEIGHT=19 ALT='('><IMG SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('
ALIGN=TOP><IMG SRC='_ca.gif' WIDTH=11 HEIGHT=19 ALT='A'> <IMG
SRC='in.gif' WIDTH=10 HEIGHT=19 ALT='e.'> <IMG SRC='bbc.gif' WIDTH=12
HEIGHT=19 ALT='CC'> <IMG SRC='wedge.gif' WIDTH=11 HEIGHT=19 ALT='/\'
ALIGN=TOP> <IMG SRC='_cb.gif' WIDTH=12 HEIGHT=19 ALT='B'> <IMG
SRC='in.gif' WIDTH=10 HEIGHT=19 ALT='e.'> <IMG SRC='scrh.gif'
WIDTH=19 HEIGHT=19 ALT='H~'> <IMG SRC='wedge.gif' WIDTH=11 HEIGHT=19
ALT='/\'> <IMG SRC='_cc.gif' WIDTH=12 HEIGHT=19 ALT='C'>
<IMG SRC='in.gif' WIDTH=10 HEIGHT=19 ALT='e.'> <IMG SRC='scrh.gif'
WIDTH=19 HEIGHT=19 ALT='H~'><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19
ALT=')'> <IMG SRC='to.gif' WIDTH=15 HEIGHT=19 ALT='->'>
<IMG SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('><IMG SRC='_ca.gif'
WIDTH=11 HEIGHT=19 ALT='A'> <IMG SRC='_cdh.gif' WIDTH=9 HEIGHT=19
ALT='.h'> <IMG SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('><IMG
SRC='_cb.gif' WIDTH=12 HEIGHT=19 ALT='B'> <IMG SRC='_pvh.gif' WIDTH=18
HEIGHT=19 ALT='+h'> <IMG SRC='_cc.gif' WIDTH=12 HEIGHT=19
ALT='C'><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19 ALT=')'><IMG
SRC='rp.gif' WIDTH=5 HEIGHT=19 ALT=')'> <IMG SRC='eq.gif' WIDTH=12
HEIGHT=19 ALT='='> <IMG SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('
ALIGN=TOP><IMG SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('><IMG
SRC='_ca.gif' WIDTH=11 HEIGHT=19 ALT='A'> <IMG SRC='_cdh.gif' WIDTH=9
HEIGHT=19 ALT='.h'> <IMG SRC='_cb.gif' WIDTH=12 HEIGHT=19 ALT='B'
ALIGN=TOP><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19 ALT=')'> <IMG
SRC='_pvh.gif' WIDTH=18 HEIGHT=19 ALT='+h'> <IMG SRC='lp.gif' WIDTH=5
HEIGHT=19 ALT='('><IMG SRC='_ca.gif' WIDTH=11 HEIGHT=19 ALT='A'
ALIGN=TOP> <IMG SRC='_cdh.gif' WIDTH=9 HEIGHT=19 ALT='.h'> <IMG
SRC='_cc.gif' WIDTH=12 HEIGHT=19 ALT='C'><IMG SRC='rp.gif' WIDTH=5
HEIGHT=19 ALT=')'><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19 ALT=')'
ALIGN=TOP><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19 ALT=')'></TD></TR>
<TR ALIGN=LEFT><TD><A HREF="ax-hvdistr2.html">ax-hvdistr2</A></TD><TD>
<IMG SRC='_vdash.gif' WIDTH=10 HEIGHT=19 ALT='|-'> <IMG SRC='lp.gif'
WIDTH=5 HEIGHT=19 ALT='('><IMG SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('
ALIGN=TOP><IMG SRC='_ca.gif' WIDTH=11 HEIGHT=19 ALT='A'> <IMG
SRC='in.gif' WIDTH=10 HEIGHT=19 ALT='e.'> <IMG SRC='bbc.gif' WIDTH=12
HEIGHT=19 ALT='CC'> <IMG SRC='wedge.gif' WIDTH=11 HEIGHT=19 ALT='/\'
ALIGN=TOP> <IMG SRC='_cb.gif' WIDTH=12 HEIGHT=19 ALT='B'> <IMG
SRC='in.gif' WIDTH=10 HEIGHT=19 ALT='e.'> <IMG SRC='bbc.gif' WIDTH=12
HEIGHT=19 ALT='CC'> <IMG SRC='wedge.gif' WIDTH=11 HEIGHT=19 ALT='/\'
ALIGN=TOP> <IMG SRC='_cc.gif' WIDTH=12 HEIGHT=19 ALT='C'> <IMG
SRC='in.gif' WIDTH=10 HEIGHT=19 ALT='e.'> <IMG SRC='scrh.gif'
WIDTH=19 HEIGHT=19 ALT='H~'><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19
ALT=')'> <IMG SRC='to.gif' WIDTH=15 HEIGHT=19 ALT='->'>
<IMG SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('><IMG SRC='lp.gif' WIDTH=5
HEIGHT=19 ALT='('><IMG SRC='_ca.gif' WIDTH=11 HEIGHT=19 ALT='A'
ALIGN=TOP> <IMG SRC='plus.gif' WIDTH=13 HEIGHT=19 ALT='+'> <IMG
SRC='_cb.gif' WIDTH=12 HEIGHT=19 ALT='B'><IMG SRC='rp.gif' WIDTH=5
HEIGHT=19 ALT=')'> <IMG SRC='_cdh.gif' WIDTH=9 HEIGHT=19 ALT='.h'
ALIGN=TOP> <IMG SRC='_cc.gif' WIDTH=12 HEIGHT=19 ALT='C'><IMG
SRC='rp.gif' WIDTH=5 HEIGHT=19 ALT=')'> <IMG SRC='eq.gif' WIDTH=12
HEIGHT=19 ALT='='> <IMG SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('
ALIGN=TOP><IMG SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('><IMG
SRC='_ca.gif' WIDTH=11 HEIGHT=19 ALT='A'> <IMG SRC='_cdh.gif' WIDTH=9
HEIGHT=19 ALT='.h'> <IMG SRC='_cc.gif' WIDTH=12 HEIGHT=19 ALT='C'
ALIGN=TOP><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19 ALT=')'> <IMG
SRC='_pvh.gif' WIDTH=18 HEIGHT=19 ALT='+h'> <IMG SRC='lp.gif' WIDTH=5
HEIGHT=19 ALT='('><IMG SRC='_cb.gif' WIDTH=12 HEIGHT=19 ALT='B'
ALIGN=TOP> <IMG SRC='_cdh.gif' WIDTH=9 HEIGHT=19 ALT='.h'> <IMG
SRC='_cc.gif' WIDTH=12 HEIGHT=19 ALT='C'><IMG SRC='rp.gif' WIDTH=5
HEIGHT=19 ALT=')'><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19 ALT=')'
ALIGN=TOP><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19 ALT=')'></TD></TR>
<TR ALIGN=LEFT><TD><A HREF="ax-hvmul0.html">ax-hvmul0</A></TD><TD>
<IMG SRC='_vdash.gif' WIDTH=10 HEIGHT=19 ALT='|-'> <IMG SRC='lp.gif'
WIDTH=5 HEIGHT=19 ALT='('><IMG SRC='_ca.gif' WIDTH=11 HEIGHT=19
ALT='A'> <IMG SRC='in.gif' WIDTH=10 HEIGHT=19 ALT='e.'>
<IMG SRC='scrh.gif' WIDTH=19 HEIGHT=19 ALT='H~'> <IMG SRC='to.gif'
WIDTH=15 HEIGHT=19 ALT='->'> <IMG SRC='lp.gif' WIDTH=5 HEIGHT=19
ALT='('><IMG SRC='0.gif' WIDTH=8 HEIGHT=19 ALT='0'> <IMG
SRC='_cdh.gif' WIDTH=9 HEIGHT=19 ALT='.h'> <IMG SRC='_ca.gif'
WIDTH=11 HEIGHT=19 ALT='A'><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19
ALT=')'> <IMG SRC='eq.gif' WIDTH=12 HEIGHT=19 ALT='='> <IMG
SRC='_0vh.gif' WIDTH=14 HEIGHT=19 ALT='0h'><IMG SRC='rp.gif' WIDTH=5
HEIGHT=19 ALT=')'></TD></TR>
<TR ALIGN=LEFT><TD><A HREF="ax-hfi.html">ax-hfi</A></TD><TD>
<IMG SRC='_vdash.gif' WIDTH=10 HEIGHT=19 ALT='|-'> <IMG
SRC='_cdih.gif' WIDTH=13 HEIGHT=19 ALT='.ih'> <IMG SRC='colon.gif'
WIDTH=4 HEIGHT=19 ALT=':'><IMG SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('
ALIGN=TOP><IMG SRC='scrh.gif' WIDTH=19 HEIGHT=19 ALT='H~'> <IMG
SRC='times.gif' WIDTH=9 HEIGHT=19 ALT='X.'> <IMG SRC='scrh.gif'
WIDTH=19 HEIGHT=19 ALT='H~'><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19
ALT=')'><IMG SRC='longrightarrow.gif' WIDTH=23 HEIGHT=19 ALT='-->'
ALIGN=TOP><IMG SRC='bbc.gif' WIDTH=12 HEIGHT=19 ALT='CC'></TD></TR>
<TR ALIGN=LEFT><TD><A HREF="ax-his1.html">ax-his1</A></TD><TD>
<IMG SRC='_vdash.gif' WIDTH=10 HEIGHT=19 ALT='|-'> <IMG SRC='lp.gif'
WIDTH=5 HEIGHT=19 ALT='('><IMG SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('
ALIGN=TOP><IMG SRC='_ca.gif' WIDTH=11 HEIGHT=19 ALT='A'> <IMG
SRC='in.gif' WIDTH=10 HEIGHT=19 ALT='e.'> <IMG SRC='scrh.gif'
WIDTH=19 HEIGHT=19 ALT='H~'> <IMG SRC='wedge.gif' WIDTH=11 HEIGHT=19
ALT='/\'> <IMG SRC='_cb.gif' WIDTH=12 HEIGHT=19 ALT='B'>
<IMG SRC='in.gif' WIDTH=10 HEIGHT=19 ALT='e.'> <IMG SRC='scrh.gif'
WIDTH=19 HEIGHT=19 ALT='H~'><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19
ALT=')'> <IMG SRC='to.gif' WIDTH=15 HEIGHT=19 ALT='->'>
<IMG SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('><IMG SRC='_ca.gif'
WIDTH=11 HEIGHT=19 ALT='A'> <IMG SRC='_cdih.gif' WIDTH=13 HEIGHT=19
ALT='.ih'> <IMG SRC='_cb.gif' WIDTH=12 HEIGHT=19 ALT='B'
ALIGN=TOP><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19 ALT=')'> <IMG
SRC='eq.gif' WIDTH=12 HEIGHT=19 ALT='='> <IMG SRC='lp.gif' WIDTH=5
HEIGHT=19 ALT='('><IMG SRC='ast.gif' WIDTH=7 HEIGHT=19 ALT='*'
ALIGN=TOP><IMG SRC='backtick.gif' WIDTH=4 HEIGHT=19 ALT='`'><IMG
SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('><IMG SRC='_cb.gif' WIDTH=12
HEIGHT=19 ALT='B'> <IMG SRC='_cdih.gif' WIDTH=13 HEIGHT=19 ALT='.ih'
ALIGN=TOP> <IMG SRC='_ca.gif' WIDTH=11 HEIGHT=19 ALT='A'><IMG
SRC='rp.gif' WIDTH=5 HEIGHT=19 ALT=')'><IMG SRC='rp.gif' WIDTH=5
HEIGHT=19 ALT=')'><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19 ALT=')'
ALIGN=TOP></TD></TR>
<TR ALIGN=LEFT><TD><A HREF="ax-his2.html">ax-his2</A></TD><TD>
<IMG SRC='_vdash.gif' WIDTH=10 HEIGHT=19 ALT='|-'> <IMG SRC='lp.gif'
WIDTH=5 HEIGHT=19 ALT='('><IMG SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('
ALIGN=TOP><IMG SRC='_ca.gif' WIDTH=11 HEIGHT=19 ALT='A'> <IMG
SRC='in.gif' WIDTH=10 HEIGHT=19 ALT='e.'> <IMG SRC='scrh.gif'
WIDTH=19 HEIGHT=19 ALT='H~'> <IMG SRC='wedge.gif' WIDTH=11 HEIGHT=19
ALT='/\'> <IMG SRC='_cb.gif' WIDTH=12 HEIGHT=19 ALT='B'>
<IMG SRC='in.gif' WIDTH=10 HEIGHT=19 ALT='e.'> <IMG SRC='scrh.gif'
WIDTH=19 HEIGHT=19 ALT='H~'> <IMG SRC='wedge.gif' WIDTH=11 HEIGHT=19
ALT='/\'> <IMG SRC='_cc.gif' WIDTH=12 HEIGHT=19 ALT='C'>
<IMG SRC='in.gif' WIDTH=10 HEIGHT=19 ALT='e.'> <IMG SRC='scrh.gif'
WIDTH=19 HEIGHT=19 ALT='H~'><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19
ALT=')'> <IMG SRC='to.gif' WIDTH=15 HEIGHT=19 ALT='->'>
<IMG SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('><IMG SRC='lp.gif' WIDTH=5
HEIGHT=19 ALT='('><IMG SRC='_ca.gif' WIDTH=11 HEIGHT=19 ALT='A'
ALIGN=TOP> <IMG SRC='_pvh.gif' WIDTH=18 HEIGHT=19 ALT='+h'> <IMG
SRC='_cb.gif' WIDTH=12 HEIGHT=19 ALT='B'><IMG SRC='rp.gif' WIDTH=5
HEIGHT=19 ALT=')'> <IMG SRC='_cdih.gif' WIDTH=13 HEIGHT=19 ALT='.ih'
ALIGN=TOP> <IMG SRC='_cc.gif' WIDTH=12 HEIGHT=19 ALT='C'><IMG
SRC='rp.gif' WIDTH=5 HEIGHT=19 ALT=')'> <IMG SRC='eq.gif' WIDTH=12
HEIGHT=19 ALT='='> <IMG SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('
ALIGN=TOP><IMG SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('><IMG
SRC='_ca.gif' WIDTH=11 HEIGHT=19 ALT='A'> <IMG SRC='_cdih.gif'
WIDTH=13 HEIGHT=19 ALT='.ih'> <IMG SRC='_cc.gif' WIDTH=12 HEIGHT=19
ALT='C'><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19 ALT=')'> <IMG
SRC='plus.gif' WIDTH=13 HEIGHT=19 ALT='+'> <IMG SRC='lp.gif' WIDTH=5
HEIGHT=19 ALT='('><IMG SRC='_cb.gif' WIDTH=12 HEIGHT=19 ALT='B'
ALIGN=TOP> <IMG SRC='_cdih.gif' WIDTH=13 HEIGHT=19 ALT='.ih'> <IMG
SRC='_cc.gif' WIDTH=12 HEIGHT=19 ALT='C'><IMG SRC='rp.gif' WIDTH=5
HEIGHT=19 ALT=')'><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19 ALT=')'
ALIGN=TOP><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19 ALT=')'></TD></TR>
<TR ALIGN=LEFT><TD><A HREF="ax-his3.html">ax-his3</A></TD><TD>
<IMG SRC='_vdash.gif' WIDTH=10 HEIGHT=19 ALT='|-'> <IMG SRC='lp.gif'
WIDTH=5 HEIGHT=19 ALT='('><IMG SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('
ALIGN=TOP><IMG SRC='_ca.gif' WIDTH=11 HEIGHT=19 ALT='A'> <IMG
SRC='in.gif' WIDTH=10 HEIGHT=19 ALT='e.'> <IMG SRC='bbc.gif' WIDTH=12
HEIGHT=19 ALT='CC'> <IMG SRC='wedge.gif' WIDTH=11 HEIGHT=19 ALT='/\'
ALIGN=TOP> <IMG SRC='_cb.gif' WIDTH=12 HEIGHT=19 ALT='B'> <IMG
SRC='in.gif' WIDTH=10 HEIGHT=19 ALT='e.'> <IMG SRC='scrh.gif'
WIDTH=19 HEIGHT=19 ALT='H~'> <IMG SRC='wedge.gif' WIDTH=11 HEIGHT=19
ALT='/\'> <IMG SRC='_cc.gif' WIDTH=12 HEIGHT=19 ALT='C'>
<IMG SRC='in.gif' WIDTH=10 HEIGHT=19 ALT='e.'> <IMG SRC='scrh.gif'
WIDTH=19 HEIGHT=19 ALT='H~'><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19
ALT=')'> <IMG SRC='to.gif' WIDTH=15 HEIGHT=19 ALT='->'>
<IMG SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('><IMG SRC='lp.gif' WIDTH=5
HEIGHT=19 ALT='('><IMG SRC='_ca.gif' WIDTH=11 HEIGHT=19 ALT='A'
ALIGN=TOP> <IMG SRC='_cdh.gif' WIDTH=9 HEIGHT=19 ALT='.h'> <IMG
SRC='_cb.gif' WIDTH=12 HEIGHT=19 ALT='B'><IMG SRC='rp.gif' WIDTH=5
HEIGHT=19 ALT=')'> <IMG SRC='_cdih.gif' WIDTH=13 HEIGHT=19 ALT='.ih'
ALIGN=TOP> <IMG SRC='_cc.gif' WIDTH=12 HEIGHT=19 ALT='C'><IMG
SRC='rp.gif' WIDTH=5 HEIGHT=19 ALT=')'> <IMG SRC='eq.gif' WIDTH=12
HEIGHT=19 ALT='='> <IMG SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('
ALIGN=TOP><IMG SRC='_ca.gif' WIDTH=11 HEIGHT=19 ALT='A'> <IMG
SRC='cdot.gif' WIDTH=4 HEIGHT=19 ALT='x.'> <IMG SRC='lp.gif' WIDTH=5
HEIGHT=19 ALT='('><IMG SRC='_cb.gif' WIDTH=12 HEIGHT=19 ALT='B'
ALIGN=TOP> <IMG SRC='_cdih.gif' WIDTH=13 HEIGHT=19 ALT='.ih'> <IMG
SRC='_cc.gif' WIDTH=12 HEIGHT=19 ALT='C'><IMG SRC='rp.gif' WIDTH=5
HEIGHT=19 ALT=')'><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19 ALT=')'
ALIGN=TOP><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19 ALT=')'></TD></TR>
<TR ALIGN=LEFT><TD><A HREF="ax-his4.html">ax-his4</A></TD><TD>
<IMG SRC='_vdash.gif' WIDTH=10 HEIGHT=19 ALT='|-'> <IMG SRC='lp.gif'
WIDTH=5 HEIGHT=19 ALT='('><IMG SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('
ALIGN=TOP><IMG SRC='_ca.gif' WIDTH=11 HEIGHT=19 ALT='A'> <IMG
SRC='in.gif' WIDTH=10 HEIGHT=19 ALT='e.'> <IMG SRC='scrh.gif'
WIDTH=19 HEIGHT=19 ALT='H~'> <IMG SRC='wedge.gif' WIDTH=11 HEIGHT=19
ALT='/\'> <IMG SRC='_ca.gif' WIDTH=11 HEIGHT=19 ALT='A'>
<IMG SRC='ne.gif' WIDTH=12 HEIGHT=19 ALT='=/='> <IMG SRC='_0vh.gif'
WIDTH=14 HEIGHT=19 ALT='0h'><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19
ALT=')'> <IMG SRC='to.gif' WIDTH=15 HEIGHT=19 ALT='->'>
<IMG SRC='0.gif' WIDTH=8 HEIGHT=19 ALT='0'> <IMG SRC='lt.gif'
WIDTH=11 HEIGHT=19 ALT='<'> <IMG SRC='lp.gif' WIDTH=5 HEIGHT=19
ALT='('><IMG SRC='_ca.gif' WIDTH=11 HEIGHT=19 ALT='A'> <IMG
SRC='_cdih.gif' WIDTH=13 HEIGHT=19 ALT='.ih'> <IMG SRC='_ca.gif'
WIDTH=11 HEIGHT=19 ALT='A'><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19
ALT=')'><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19 ALT=')'
ALIGN=TOP></TD></TR>
<TR ALIGN=LEFT><TD><A HREF="ax-hcompl.html">ax-hcompl</A></TD><TD>
<IMG SRC='_vdash.gif' WIDTH=10 HEIGHT=19 ALT='|-'> <IMG SRC='lp.gif'
WIDTH=5 HEIGHT=19 ALT='('><IMG SRC='_cf.gif' WIDTH=13 HEIGHT=19
ALT='F'> <IMG SRC='in.gif' WIDTH=10 HEIGHT=19 ALT='e.'>
<IMG SRC='_cauchy.gif' WIDTH=47 HEIGHT=19 ALT='Cauchy'> <IMG
SRC='to.gif' WIDTH=15 HEIGHT=19 ALT='->'> <IMG SRC='exists.gif'
WIDTH=9 HEIGHT=19 ALT='E.'><IMG SRC='_x.gif' WIDTH=10 HEIGHT=19
ALT='x'> <IMG SRC='in.gif' WIDTH=10 HEIGHT=19 ALT='e.'>
<IMG SRC='scrh.gif' WIDTH=19 HEIGHT=19 ALT='H~'><IMG SRC='_cf.gif'
WIDTH=13 HEIGHT=19 ALT='F'> <IMG SRC='_squigv.gif' WIDTH=21 HEIGHT=19
ALT='~~>v'> <IMG SRC='_x.gif' WIDTH=10 HEIGHT=19 ALT='x'
ALIGN=TOP><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19 ALT=')'></TD></TR>
</TABLE></CENTER>
<P><I>Comments on the axioms.</I> The first axiom just says that the
primitive class
<IMG SRC='scrh.gif' WIDTH=19 HEIGHT=19 ALT='H~'>
exists (is a member of the universe of sets <I>V</I>).
The next 11 axioms are the axioms for any vector space with
an unspecified dimension; they are the same as those you would find in
any linear algebra book, except for the notation and possibly their
precise form.
<P>The next 5 axioms show the properties of the special inner product
<IMG SRC='_cdih.gif' WIDTH=13 HEIGHT=19 ALT='.ih'>. The official
name for this inner product is a "sesquilinear Hermitian
mapping". (Sesquilinear means "one-and-a-half linear," i.e.,
antilinear in the first argument and linear in the second.) The symbol
<IMG SRC='ast.gif' WIDTH=7 HEIGHT=19 ALT='*'> in Axiom <A
HREF="ax-his1.html">ax-his1</A> is the complex conjugate (<A
HREF="cjval.html">cjval</A>). See <A
HREF="mmset.html#function">Notation for Function Values</A> for an
explanation of why we use this notation rather than the standard
superscript asterisk used in textbooks; this will help you understand
some of our other non-standard notation as well.
<P>The last axiom, which is the most important and also the most
complicated, is called the Completeness Axiom, and is shown above using
abbreviations. You can click on its links to expand the abbreviations.
It basically says that the limit of any converging ("Cauchy")
sequence of vectors in Hilbert space converges to a vector in Hilbert
space. To understand what completeness means, consider this analogy:
the sequence 3, 3.1, 3.14, 3.1415, 3.14159... converges to pi. This is
a converging sequence of rational numbers, but it converges to something
that is not a rational number, meaning the set of rational numbers is
<I>not</I> complete. The set of real numbers, on the other hand,
<I>is</I> complete, because all converging sequences of real numbers
converge to a real number.
<P><HR NOSHADE SIZE=1><A NAME="definitions"></A><B><FONT
COLOR="#006633">Some Definitions</FONT></B> Here we
show explicitly a few of the definitions you will encounter in our
Hilbert space proofs. The complete list is given at the top of this
page. We typically define new symbols as a self-contained objects, which
can make their definitions seem unnecessarily complicated, but usually
their Descriptions point to simpler theorems showing their values or
other properties. For example, the vector subtraction operation <IMG
SRC='_mvh.gif' WIDTH=16 HEIGHT=19 ALT='-h'> is formally a set
of ordered pairs as shown below, but its value is just <IMG
SRC='_ca.gif' WIDTH=11 HEIGHT=19 ALT='A'> <IMG SRC='_pvh.gif'
WIDTH=18 HEIGHT=19 ALT='+h'> <IMG SRC='lp.gif' WIDTH=5
HEIGHT=19 ALT='('><IMG SRC='shortminus.gif' WIDTH=8 HEIGHT=19
ALT='-u'><IMG SRC='1.gif' WIDTH=7 HEIGHT=19 ALT='1'>
<IMG SRC='_cdh.gif' WIDTH=9 HEIGHT=19 ALT='.h'> <IMG
SRC='_cb.gif' WIDTH=12 HEIGHT=19 ALT='B'><IMG SRC='rp.gif'
WIDTH=5 HEIGHT=19 ALT=')'> as can be seen from theorem <A
HREF="hvsubval.html">hvsubval</A>.
<P><CENTER><TABLE BORDER CELLSPACING=0 BGCOLOR="#EEFFFA">
<CAPTION><B>Some Definitions for Hilbert Space</B></CAPTION>
<TR ALIGN=LEFT><TD>The vector subtraction operation
<A HREF="df-hvsub.html">df-hvsub</A></TD><TD>
<IMG SRC='_vdash.gif' WIDTH=10 HEIGHT=19 ALT='|-'> <IMG
SRC='_mvh.gif' WIDTH=16 HEIGHT=19 ALT='-h'> <IMG SRC='eq.gif'
WIDTH=12 HEIGHT=19 ALT='='> <IMG SRC='lbrace.gif' WIDTH=6 HEIGHT=19
ALT='{'><IMG SRC='langle.gif' WIDTH=4 HEIGHT=19 ALT='<.'
ALIGN=TOP><IMG SRC='langle.gif' WIDTH=4 HEIGHT=19 ALT='<.'><IMG
SRC='_x.gif' WIDTH=10 HEIGHT=19 ALT='x'><IMG SRC='comma.gif' WIDTH=4
HEIGHT=19 ALT=','><IMG SRC='_y.gif' WIDTH=9 HEIGHT=19 ALT='y'
ALIGN=TOP><IMG SRC='rangle.gif' WIDTH=4 HEIGHT=19 ALT='>.'><IMG
SRC='comma.gif' WIDTH=4 HEIGHT=19 ALT=','><IMG SRC='_z.gif' WIDTH=9
HEIGHT=19 ALT='z'><IMG SRC='rangle.gif' WIDTH=4 HEIGHT=19 ALT='>.'
ALIGN=TOP> <IMG SRC='vert.gif' WIDTH=3 HEIGHT=19 ALT='|'> <IMG
SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('><IMG SRC='lp.gif' WIDTH=5
HEIGHT=19 ALT='('><IMG SRC='_x.gif' WIDTH=10 HEIGHT=19 ALT='x'
ALIGN=TOP> <IMG SRC='in.gif' WIDTH=10 HEIGHT=19 ALT='e.'> <IMG
SRC='scrh.gif' WIDTH=19 HEIGHT=19 ALT='H~'> <IMG SRC='wedge.gif'
WIDTH=11 HEIGHT=19 ALT='/\'> <IMG SRC='_y.gif' WIDTH=9 HEIGHT=19
ALT='y'> <IMG SRC='in.gif' WIDTH=10 HEIGHT=19 ALT='e.'>
<IMG SRC='scrh.gif' WIDTH=19 HEIGHT=19 ALT='H~'><IMG SRC='rp.gif'
WIDTH=5 HEIGHT=19 ALT=')'> <IMG SRC='wedge.gif' WIDTH=11 HEIGHT=19
ALT='/\'> <IMG SRC='_z.gif' WIDTH=9 HEIGHT=19 ALT='z'> <IMG
SRC='eq.gif' WIDTH=12 HEIGHT=19 ALT='='> <IMG SRC='lp.gif' WIDTH=5
HEIGHT=19 ALT='('><IMG SRC='_x.gif' WIDTH=10 HEIGHT=19 ALT='x'
ALIGN=TOP> <IMG SRC='_pvh.gif' WIDTH=18 HEIGHT=19 ALT='+h'> <IMG
SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('><IMG SRC='shortminus.gif'
WIDTH=8 HEIGHT=19 ALT='-u'><IMG SRC='1.gif' WIDTH=7 HEIGHT=19 ALT='1'
ALIGN=TOP> <IMG SRC='_cdh.gif' WIDTH=9 HEIGHT=19 ALT='.h'> <IMG
SRC='_y.gif' WIDTH=9 HEIGHT=19 ALT='y'><IMG SRC='rp.gif' WIDTH=5
HEIGHT=19 ALT=')'><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19 ALT=')'
ALIGN=TOP><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19 ALT=')'><IMG
SRC='rbrace.gif' WIDTH=6 HEIGHT=19 ALT='}'></TD></TR>
<TR ALIGN=LEFT><TD>The norm of a vector
<A HREF="df-hnorm.html">df-hnorm</A></TD><TD>
<IMG SRC='_vdash.gif' WIDTH=10 HEIGHT=19 ALT='|-'> <IMG
SRC='_normh.gif' WIDTH=38 HEIGHT=19 ALT='normh'> <IMG SRC='eq.gif'
WIDTH=12 HEIGHT=19 ALT='='> <IMG SRC='lbrace.gif' WIDTH=6 HEIGHT=19
ALT='{'><IMG SRC='langle.gif' WIDTH=4 HEIGHT=19 ALT='<.'
ALIGN=TOP><IMG SRC='_x.gif' WIDTH=10 HEIGHT=19 ALT='x'><IMG
SRC='comma.gif' WIDTH=4 HEIGHT=19 ALT=','><IMG SRC='_y.gif' WIDTH=9
HEIGHT=19 ALT='y'><IMG SRC='rangle.gif' WIDTH=4 HEIGHT=19 ALT='>.'
ALIGN=TOP> <IMG SRC='vert.gif' WIDTH=3 HEIGHT=19 ALT='|'> <IMG
SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('><IMG SRC='_x.gif' WIDTH=10
HEIGHT=19 ALT='x'> <IMG SRC='in.gif' WIDTH=10 HEIGHT=19 ALT='e.'
ALIGN=TOP> <IMG SRC='scrh.gif' WIDTH=19 HEIGHT=19 ALT='H~'> <IMG
SRC='wedge.gif' WIDTH=11 HEIGHT=19 ALT='/\'> <IMG SRC='_y.gif'
WIDTH=9 HEIGHT=19 ALT='y'> <IMG SRC='eq.gif' WIDTH=12 HEIGHT=19
ALT='='> <IMG SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('><IMG
SRC='surd.gif' WIDTH=14 HEIGHT=19 ALT='sqrt'><IMG SRC='backtick.gif'
WIDTH=4 HEIGHT=19 ALT='`'><IMG SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('
ALIGN=TOP><IMG SRC='_x.gif' WIDTH=10 HEIGHT=19 ALT='x'> <IMG
SRC='_cdih.gif' WIDTH=13 HEIGHT=19 ALT='.ih'> <IMG SRC='_x.gif'
WIDTH=10 HEIGHT=19 ALT='x'><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19
ALT=')'><IMG SRC='rp.gif' WIDTH=5 HEIGHT=19 ALT=')'><IMG
SRC='rp.gif' WIDTH=5 HEIGHT=19 ALT=')'><IMG SRC='rbrace.gif' WIDTH=6
HEIGHT=19 ALT='}'></TD></TR>
<TR ALIGN=LEFT><TD>The set of all Cauchy sequences
<A HREF="df-hcau.html">df-hcau</A></TD><TD>
<IMG SRC='_vdash.gif' WIDTH=10 HEIGHT=19 ALT='|-'> <IMG
SRC='_cauchy.gif' WIDTH=47 HEIGHT=19 ALT='Cauchy'> <IMG SRC='eq.gif'
WIDTH=12 HEIGHT=19 ALT='='> <IMG SRC='lbrace.gif' WIDTH=6 HEIGHT=19
ALT='{'><IMG SRC='_f.gif' WIDTH=9 HEIGHT=19 ALT='f'> <IMG
SRC='vert.gif' WIDTH=3 HEIGHT=19 ALT='|'> <IMG SRC='lp.gif' WIDTH=5
HEIGHT=19 ALT='('><IMG SRC='_f.gif' WIDTH=9 HEIGHT=19 ALT='f'
ALIGN=TOP><IMG SRC='colon.gif' WIDTH=4 HEIGHT=19 ALT=':'><IMG
SRC='bbn.gif' WIDTH=12 HEIGHT=19 ALT='NN'><IMG
SRC='longrightarrow.gif' WIDTH=23 HEIGHT=19 ALT='-->'><IMG
SRC='scrh.gif' WIDTH=19 HEIGHT=19 ALT='H~'> <IMG SRC='wedge.gif'
WIDTH=11 HEIGHT=19 ALT='/\'> <IMG SRC='forall.gif' WIDTH=10 HEIGHT=19
ALT='A.'><IMG SRC='_x.gif' WIDTH=10 HEIGHT=19 ALT='x'> <IMG
SRC='in.gif' WIDTH=10 HEIGHT=19 ALT='e.'> <IMG SRC='bbr.gif' WIDTH=13
HEIGHT=19 ALT='RR'><IMG SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='('
ALIGN=TOP><IMG SRC='0.gif' WIDTH=8 HEIGHT=19 ALT='0'> <IMG
SRC='lt.gif' WIDTH=11 HEIGHT=19 ALT='<'> <IMG SRC='_x.gif'
WIDTH=10 HEIGHT=19 ALT='x'> <IMG SRC='to.gif' WIDTH=15 HEIGHT=19
ALT='->'> <IMG SRC='exists.gif' WIDTH=9 HEIGHT=19 ALT='E.'
ALIGN=TOP><IMG SRC='_y.gif' WIDTH=9 HEIGHT=19 ALT='y'> <IMG
SRC='in.gif' WIDTH=10 HEIGHT=19 ALT='e.'> <IMG SRC='bbn.gif' WIDTH=12
HEIGHT=19 ALT='NN'><IMG SRC='forall.gif' WIDTH=10 HEIGHT=19 ALT='A.'
ALIGN=TOP><IMG SRC='_z.gif' WIDTH=9 HEIGHT=19 ALT='z'> <IMG
SRC='in.gif' WIDTH=10 HEIGHT=19 ALT='e.'> <IMG SRC='bbn.gif' WIDTH=12