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Mirrors > Home > HSE Home > Th. List > hhnmoi | Structured version Visualization version GIF version |
Description: The norm of an operator in Hilbert space. (Contributed by NM, 19-Nov-2007.) (Revised by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hhnmo.1 | ⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 |
hhnmo.2 | ⊢ 𝑁 = (𝑈 normOpOLD 𝑈) |
Ref | Expression |
---|---|
hhnmoi | ⊢ normop = 𝑁 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nmop 29774 | . 2 ⊢ normop = (𝑡 ∈ ( ℋ ↑m ℋ) ↦ sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑡‘𝑦)))}, ℝ*, < )) | |
2 | hhnmo.1 | . . . 4 ⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 | |
3 | 2 | hhnv 29100 | . . 3 ⊢ 𝑈 ∈ NrmCVec |
4 | 2 | hhba 29102 | . . . 4 ⊢ ℋ = (BaseSet‘𝑈) |
5 | 2 | hhnm 29106 | . . . 4 ⊢ normℎ = (normCV‘𝑈) |
6 | hhnmo.2 | . . . 4 ⊢ 𝑁 = (𝑈 normOpOLD 𝑈) | |
7 | 4, 4, 5, 5, 6 | nmoofval 28697 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑈 ∈ NrmCVec) → 𝑁 = (𝑡 ∈ ( ℋ ↑m ℋ) ↦ sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑡‘𝑦)))}, ℝ*, < ))) |
8 | 3, 3, 7 | mp2an 692 | . 2 ⊢ 𝑁 = (𝑡 ∈ ( ℋ ↑m ℋ) ↦ sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑡‘𝑦)))}, ℝ*, < )) |
9 | 1, 8 | eqtr4i 2764 | 1 ⊢ normop = 𝑁 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 = wceq 1542 ∈ wcel 2114 {cab 2716 ∃wrex 3054 〈cop 4522 class class class wbr 5030 ↦ cmpt 5110 ‘cfv 6339 (class class class)co 7170 ↑m cmap 8437 supcsup 8977 1c1 10616 ℝ*cxr 10752 < clt 10753 ≤ cle 10754 NrmCVeccnv 28519 normOpOLD cnmoo 28676 ℋchba 28854 +ℎ cva 28855 ·ℎ csm 28856 normℎcno 28858 normopcnop 28880 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 ax-pre-sup 10693 ax-hilex 28934 ax-hfvadd 28935 ax-hvcom 28936 ax-hvass 28937 ax-hv0cl 28938 ax-hvaddid 28939 ax-hfvmul 28940 ax-hvmulid 28941 ax-hvmulass 28942 ax-hvdistr1 28943 ax-hvdistr2 28944 ax-hvmul0 28945 ax-hfi 29014 ax-his1 29017 ax-his2 29018 ax-his3 29019 ax-his4 29020 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-1st 7714 df-2nd 7715 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-er 8320 df-en 8556 df-dom 8557 df-sdom 8558 df-sup 8979 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-div 11376 df-nn 11717 df-2 11779 df-3 11780 df-4 11781 df-n0 11977 df-z 12063 df-uz 12325 df-rp 12473 df-seq 13461 df-exp 13522 df-cj 14548 df-re 14549 df-im 14550 df-sqrt 14684 df-abs 14685 df-grpo 28428 df-gid 28429 df-ablo 28480 df-vc 28494 df-nv 28527 df-va 28530 df-ba 28531 df-nmcv 28535 df-nmoo 28680 df-hnorm 28903 df-hvsub 28906 df-nmop 29774 |
This theorem is referenced by: hhbloi 29837 nmopub2tHIL 29845 nmlnop0iHIL 29931 |
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