![]() |
Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > HSE Home > Th. List > riesz1 | Structured version Visualization version GIF version |
Description: Part 1 of the Riesz representation theorem for bounded linear functionals. A linear functional is bounded iff its value can be expressed as an inner product. Part of Theorem 17.3 of [Halmos] p. 31. For part 2, see riesz2 29849. For the continuous linear functional version, see riesz3i 29845 and riesz4 29847. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
riesz1 | ⊢ (𝑇 ∈ LinFn → ((normfn‘𝑇) ∈ ℝ ↔ ∃𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑇‘𝑥) = (𝑥 ·ih 𝑦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lnfncnbd 29840 | . 2 ⊢ (𝑇 ∈ LinFn → (𝑇 ∈ ContFn ↔ (normfn‘𝑇) ∈ ℝ)) | |
2 | elin 3897 | . . . . 5 ⊢ (𝑇 ∈ (LinFn ∩ ContFn) ↔ (𝑇 ∈ LinFn ∧ 𝑇 ∈ ContFn)) | |
3 | fveq1 6644 | . . . . . . . 8 ⊢ (𝑇 = if(𝑇 ∈ (LinFn ∩ ContFn), 𝑇, ( ℋ × {0})) → (𝑇‘𝑥) = (if(𝑇 ∈ (LinFn ∩ ContFn), 𝑇, ( ℋ × {0}))‘𝑥)) | |
4 | 3 | eqeq1d 2800 | . . . . . . 7 ⊢ (𝑇 = if(𝑇 ∈ (LinFn ∩ ContFn), 𝑇, ( ℋ × {0})) → ((𝑇‘𝑥) = (𝑥 ·ih 𝑦) ↔ (if(𝑇 ∈ (LinFn ∩ ContFn), 𝑇, ( ℋ × {0}))‘𝑥) = (𝑥 ·ih 𝑦))) |
5 | 4 | rexralbidv 3260 | . . . . . 6 ⊢ (𝑇 = if(𝑇 ∈ (LinFn ∩ ContFn), 𝑇, ( ℋ × {0})) → (∃𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑇‘𝑥) = (𝑥 ·ih 𝑦) ↔ ∃𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (if(𝑇 ∈ (LinFn ∩ ContFn), 𝑇, ( ℋ × {0}))‘𝑥) = (𝑥 ·ih 𝑦))) |
6 | inss1 4155 | . . . . . . . 8 ⊢ (LinFn ∩ ContFn) ⊆ LinFn | |
7 | 0lnfn 29768 | . . . . . . . . . 10 ⊢ ( ℋ × {0}) ∈ LinFn | |
8 | 0cnfn 29763 | . . . . . . . . . 10 ⊢ ( ℋ × {0}) ∈ ContFn | |
9 | elin 3897 | . . . . . . . . . 10 ⊢ (( ℋ × {0}) ∈ (LinFn ∩ ContFn) ↔ (( ℋ × {0}) ∈ LinFn ∧ ( ℋ × {0}) ∈ ContFn)) | |
10 | 7, 8, 9 | mpbir2an 710 | . . . . . . . . 9 ⊢ ( ℋ × {0}) ∈ (LinFn ∩ ContFn) |
11 | 10 | elimel 4492 | . . . . . . . 8 ⊢ if(𝑇 ∈ (LinFn ∩ ContFn), 𝑇, ( ℋ × {0})) ∈ (LinFn ∩ ContFn) |
12 | 6, 11 | sselii 3912 | . . . . . . 7 ⊢ if(𝑇 ∈ (LinFn ∩ ContFn), 𝑇, ( ℋ × {0})) ∈ LinFn |
13 | inss2 4156 | . . . . . . . 8 ⊢ (LinFn ∩ ContFn) ⊆ ContFn | |
14 | 13, 11 | sselii 3912 | . . . . . . 7 ⊢ if(𝑇 ∈ (LinFn ∩ ContFn), 𝑇, ( ℋ × {0})) ∈ ContFn |
15 | 12, 14 | riesz3i 29845 | . . . . . 6 ⊢ ∃𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (if(𝑇 ∈ (LinFn ∩ ContFn), 𝑇, ( ℋ × {0}))‘𝑥) = (𝑥 ·ih 𝑦) |
16 | 5, 15 | dedth 4481 | . . . . 5 ⊢ (𝑇 ∈ (LinFn ∩ ContFn) → ∃𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑇‘𝑥) = (𝑥 ·ih 𝑦)) |
17 | 2, 16 | sylbir 238 | . . . 4 ⊢ ((𝑇 ∈ LinFn ∧ 𝑇 ∈ ContFn) → ∃𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑇‘𝑥) = (𝑥 ·ih 𝑦)) |
18 | 17 | ex 416 | . . 3 ⊢ (𝑇 ∈ LinFn → (𝑇 ∈ ContFn → ∃𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑇‘𝑥) = (𝑥 ·ih 𝑦))) |
19 | normcl 28908 | . . . . . . 7 ⊢ (𝑦 ∈ ℋ → (normℎ‘𝑦) ∈ ℝ) | |
20 | 19 | adantl 485 | . . . . . 6 ⊢ ((𝑇 ∈ LinFn ∧ 𝑦 ∈ ℋ) → (normℎ‘𝑦) ∈ ℝ) |
21 | fveq2 6645 | . . . . . . . . . . 11 ⊢ ((𝑇‘𝑥) = (𝑥 ·ih 𝑦) → (abs‘(𝑇‘𝑥)) = (abs‘(𝑥 ·ih 𝑦))) | |
22 | 21 | adantl 485 | . . . . . . . . . 10 ⊢ ((((𝑇 ∈ LinFn ∧ 𝑥 ∈ ℋ) ∧ 𝑦 ∈ ℋ) ∧ (𝑇‘𝑥) = (𝑥 ·ih 𝑦)) → (abs‘(𝑇‘𝑥)) = (abs‘(𝑥 ·ih 𝑦))) |
23 | bcs 28964 | . . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (abs‘(𝑥 ·ih 𝑦)) ≤ ((normℎ‘𝑥) · (normℎ‘𝑦))) | |
24 | normcl 28908 | . . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℋ → (normℎ‘𝑥) ∈ ℝ) | |
25 | recn 10616 | . . . . . . . . . . . . . . 15 ⊢ ((normℎ‘𝑥) ∈ ℝ → (normℎ‘𝑥) ∈ ℂ) | |
26 | recn 10616 | . . . . . . . . . . . . . . 15 ⊢ ((normℎ‘𝑦) ∈ ℝ → (normℎ‘𝑦) ∈ ℂ) | |
27 | mulcom 10612 | . . . . . . . . . . . . . . 15 ⊢ (((normℎ‘𝑥) ∈ ℂ ∧ (normℎ‘𝑦) ∈ ℂ) → ((normℎ‘𝑥) · (normℎ‘𝑦)) = ((normℎ‘𝑦) · (normℎ‘𝑥))) | |
28 | 25, 26, 27 | syl2an 598 | . . . . . . . . . . . . . 14 ⊢ (((normℎ‘𝑥) ∈ ℝ ∧ (normℎ‘𝑦) ∈ ℝ) → ((normℎ‘𝑥) · (normℎ‘𝑦)) = ((normℎ‘𝑦) · (normℎ‘𝑥))) |
29 | 24, 19, 28 | syl2an 598 | . . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((normℎ‘𝑥) · (normℎ‘𝑦)) = ((normℎ‘𝑦) · (normℎ‘𝑥))) |
30 | 23, 29 | breqtrd 5056 | . . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (abs‘(𝑥 ·ih 𝑦)) ≤ ((normℎ‘𝑦) · (normℎ‘𝑥))) |
31 | 30 | adantll 713 | . . . . . . . . . . 11 ⊢ (((𝑇 ∈ LinFn ∧ 𝑥 ∈ ℋ) ∧ 𝑦 ∈ ℋ) → (abs‘(𝑥 ·ih 𝑦)) ≤ ((normℎ‘𝑦) · (normℎ‘𝑥))) |
32 | 31 | adantr 484 | . . . . . . . . . 10 ⊢ ((((𝑇 ∈ LinFn ∧ 𝑥 ∈ ℋ) ∧ 𝑦 ∈ ℋ) ∧ (𝑇‘𝑥) = (𝑥 ·ih 𝑦)) → (abs‘(𝑥 ·ih 𝑦)) ≤ ((normℎ‘𝑦) · (normℎ‘𝑥))) |
33 | 22, 32 | eqbrtrd 5052 | . . . . . . . . 9 ⊢ ((((𝑇 ∈ LinFn ∧ 𝑥 ∈ ℋ) ∧ 𝑦 ∈ ℋ) ∧ (𝑇‘𝑥) = (𝑥 ·ih 𝑦)) → (abs‘(𝑇‘𝑥)) ≤ ((normℎ‘𝑦) · (normℎ‘𝑥))) |
34 | 33 | ex 416 | . . . . . . . 8 ⊢ (((𝑇 ∈ LinFn ∧ 𝑥 ∈ ℋ) ∧ 𝑦 ∈ ℋ) → ((𝑇‘𝑥) = (𝑥 ·ih 𝑦) → (abs‘(𝑇‘𝑥)) ≤ ((normℎ‘𝑦) · (normℎ‘𝑥)))) |
35 | 34 | an32s 651 | . . . . . . 7 ⊢ (((𝑇 ∈ LinFn ∧ 𝑦 ∈ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝑇‘𝑥) = (𝑥 ·ih 𝑦) → (abs‘(𝑇‘𝑥)) ≤ ((normℎ‘𝑦) · (normℎ‘𝑥)))) |
36 | 35 | ralimdva 3144 | . . . . . 6 ⊢ ((𝑇 ∈ LinFn ∧ 𝑦 ∈ ℋ) → (∀𝑥 ∈ ℋ (𝑇‘𝑥) = (𝑥 ·ih 𝑦) → ∀𝑥 ∈ ℋ (abs‘(𝑇‘𝑥)) ≤ ((normℎ‘𝑦) · (normℎ‘𝑥)))) |
37 | oveq1 7142 | . . . . . . . . 9 ⊢ (𝑧 = (normℎ‘𝑦) → (𝑧 · (normℎ‘𝑥)) = ((normℎ‘𝑦) · (normℎ‘𝑥))) | |
38 | 37 | breq2d 5042 | . . . . . . . 8 ⊢ (𝑧 = (normℎ‘𝑦) → ((abs‘(𝑇‘𝑥)) ≤ (𝑧 · (normℎ‘𝑥)) ↔ (abs‘(𝑇‘𝑥)) ≤ ((normℎ‘𝑦) · (normℎ‘𝑥)))) |
39 | 38 | ralbidv 3162 | . . . . . . 7 ⊢ (𝑧 = (normℎ‘𝑦) → (∀𝑥 ∈ ℋ (abs‘(𝑇‘𝑥)) ≤ (𝑧 · (normℎ‘𝑥)) ↔ ∀𝑥 ∈ ℋ (abs‘(𝑇‘𝑥)) ≤ ((normℎ‘𝑦) · (normℎ‘𝑥)))) |
40 | 39 | rspcev 3571 | . . . . . 6 ⊢ (((normℎ‘𝑦) ∈ ℝ ∧ ∀𝑥 ∈ ℋ (abs‘(𝑇‘𝑥)) ≤ ((normℎ‘𝑦) · (normℎ‘𝑥))) → ∃𝑧 ∈ ℝ ∀𝑥 ∈ ℋ (abs‘(𝑇‘𝑥)) ≤ (𝑧 · (normℎ‘𝑥))) |
41 | 20, 36, 40 | syl6an 683 | . . . . 5 ⊢ ((𝑇 ∈ LinFn ∧ 𝑦 ∈ ℋ) → (∀𝑥 ∈ ℋ (𝑇‘𝑥) = (𝑥 ·ih 𝑦) → ∃𝑧 ∈ ℝ ∀𝑥 ∈ ℋ (abs‘(𝑇‘𝑥)) ≤ (𝑧 · (normℎ‘𝑥)))) |
42 | 41 | rexlimdva 3243 | . . . 4 ⊢ (𝑇 ∈ LinFn → (∃𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑇‘𝑥) = (𝑥 ·ih 𝑦) → ∃𝑧 ∈ ℝ ∀𝑥 ∈ ℋ (abs‘(𝑇‘𝑥)) ≤ (𝑧 · (normℎ‘𝑥)))) |
43 | lnfncon 29839 | . . . 4 ⊢ (𝑇 ∈ LinFn → (𝑇 ∈ ContFn ↔ ∃𝑧 ∈ ℝ ∀𝑥 ∈ ℋ (abs‘(𝑇‘𝑥)) ≤ (𝑧 · (normℎ‘𝑥)))) | |
44 | 42, 43 | sylibrd 262 | . . 3 ⊢ (𝑇 ∈ LinFn → (∃𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑇‘𝑥) = (𝑥 ·ih 𝑦) → 𝑇 ∈ ContFn)) |
45 | 18, 44 | impbid 215 | . 2 ⊢ (𝑇 ∈ LinFn → (𝑇 ∈ ContFn ↔ ∃𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑇‘𝑥) = (𝑥 ·ih 𝑦))) |
46 | 1, 45 | bitr3d 284 | 1 ⊢ (𝑇 ∈ LinFn → ((normfn‘𝑇) ∈ ℝ ↔ ∃𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑇‘𝑥) = (𝑥 ·ih 𝑦))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ∃wrex 3107 ∩ cin 3880 ifcif 4425 {csn 4525 class class class wbr 5030 × cxp 5517 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 ℝcr 10525 0cc0 10526 · cmul 10531 ≤ cle 10665 abscabs 14585 ℋchba 28702 ·ih csp 28705 normℎcno 28706 normfncnmf 28734 ContFnccnfn 28736 LinFnclf 28737 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cc 9846 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606 ax-hilex 28782 ax-hfvadd 28783 ax-hvcom 28784 ax-hvass 28785 ax-hv0cl 28786 ax-hvaddid 28787 ax-hfvmul 28788 ax-hvmulid 28789 ax-hvmulass 28790 ax-hvdistr1 28791 ax-hvdistr2 28792 ax-hvmul0 28793 ax-hfi 28862 ax-his1 28865 ax-his2 28866 ax-his3 28867 ax-his4 28868 ax-hcompl 28985 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-omul 8090 df-er 8272 df-map 8391 df-pm 8392 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-fi 8859 df-sup 8890 df-inf 8891 df-oi 8958 df-card 9352 df-acn 9355 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-ico 12732 df-icc 12733 df-fz 12886 df-fzo 13029 df-fl 13157 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-rlim 14838 df-sum 15035 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-rest 16688 df-topn 16689 df-0g 16707 df-gsum 16708 df-topgen 16709 df-pt 16710 df-prds 16713 df-xrs 16767 df-qtop 16772 df-imas 16773 df-xps 16775 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-mulg 18217 df-cntz 18439 df-cmn 18900 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-fbas 20088 df-fg 20089 df-cnfld 20092 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 df-cld 21624 df-ntr 21625 df-cls 21626 df-nei 21703 df-cn 21832 df-cnp 21833 df-lm 21834 df-t1 21919 df-haus 21920 df-tx 22167 df-hmeo 22360 df-fil 22451 df-fm 22543 df-flim 22544 df-flf 22545 df-xms 22927 df-ms 22928 df-tms 22929 df-cfil 23859 df-cau 23860 df-cmet 23861 df-grpo 28276 df-gid 28277 df-ginv 28278 df-gdiv 28279 df-ablo 28328 df-vc 28342 df-nv 28375 df-va 28378 df-ba 28379 df-sm 28380 df-0v 28381 df-vs 28382 df-nmcv 28383 df-ims 28384 df-dip 28484 df-ssp 28505 df-ph 28596 df-cbn 28646 df-hnorm 28751 df-hba 28752 df-hvsub 28754 df-hlim 28755 df-hcau 28756 df-sh 28990 df-ch 29004 df-oc 29035 df-ch0 29036 df-nmfn 29628 df-nlfn 29629 df-cnfn 29630 df-lnfn 29631 |
This theorem is referenced by: rnbra 29890 |
Copyright terms: Public domain | W3C validator |