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Mirrors > Home > HSE Home > Th. List > dfhnorm2 | Structured version Visualization version GIF version |
Description: Alternate definition of the norm of a vector of Hilbert space. Definition of norm in [Beran] p. 96. (Contributed by NM, 6-Jun-2008.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dfhnorm2 | ⊢ normℎ = (𝑥 ∈ ℋ ↦ (√‘(𝑥 ·ih 𝑥))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-hnorm 28672 | . 2 ⊢ normℎ = (𝑥 ∈ dom dom ·ih ↦ (√‘(𝑥 ·ih 𝑥))) | |
2 | ax-hfi 28783 | . . . . . 6 ⊢ ·ih :( ℋ × ℋ)⟶ℂ | |
3 | 2 | fdmi 6517 | . . . . 5 ⊢ dom ·ih = ( ℋ × ℋ) |
4 | 3 | dmeqi 5766 | . . . 4 ⊢ dom dom ·ih = dom ( ℋ × ℋ) |
5 | dmxpid 5793 | . . . 4 ⊢ dom ( ℋ × ℋ) = ℋ | |
6 | 4, 5 | eqtr2i 2842 | . . 3 ⊢ ℋ = dom dom ·ih |
7 | 6 | mpteq1i 5147 | . 2 ⊢ (𝑥 ∈ ℋ ↦ (√‘(𝑥 ·ih 𝑥))) = (𝑥 ∈ dom dom ·ih ↦ (√‘(𝑥 ·ih 𝑥))) |
8 | 1, 7 | eqtr4i 2844 | 1 ⊢ normℎ = (𝑥 ∈ ℋ ↦ (√‘(𝑥 ·ih 𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ↦ cmpt 5137 × cxp 5546 dom cdm 5548 ‘cfv 6348 (class class class)co 7145 ℂcc 10523 √csqrt 14580 ℋchba 28623 ·ih csp 28626 normℎcno 28627 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 ax-hfi 28783 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-mpt 5138 df-xp 5554 df-dm 5558 df-fn 6351 df-f 6352 df-hnorm 28672 |
This theorem is referenced by: normf 28827 normval 28828 hilnormi 28867 |
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