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Mirrors > Home > HSE Home > Th. List > riesz4 | Structured version Visualization version GIF version |
Description: A continuous linear functional can be expressed as an inner product. Uniqueness part of Theorem 3.9 of [Beran] p. 104. See riesz2 31753 for the bounded linear functional version. (Contributed by NM, 16-Feb-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
riesz4 | ⊢ (𝑇 ∈ (LinFn ∩ ContFn) → ∃!𝑤 ∈ ℋ ∀𝑣 ∈ ℋ (𝑇‘𝑣) = (𝑣 ·ih 𝑤)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq1 6890 | . . . . 5 ⊢ (𝑇 = if(𝑇 ∈ (LinFn ∩ ContFn), 𝑇, ( ℋ × {0})) → (𝑇‘𝑣) = (if(𝑇 ∈ (LinFn ∩ ContFn), 𝑇, ( ℋ × {0}))‘𝑣)) | |
2 | 1 | eqeq1d 2733 | . . . 4 ⊢ (𝑇 = if(𝑇 ∈ (LinFn ∩ ContFn), 𝑇, ( ℋ × {0})) → ((𝑇‘𝑣) = (𝑣 ·ih 𝑤) ↔ (if(𝑇 ∈ (LinFn ∩ ContFn), 𝑇, ( ℋ × {0}))‘𝑣) = (𝑣 ·ih 𝑤))) |
3 | 2 | ralbidv 3176 | . . 3 ⊢ (𝑇 = if(𝑇 ∈ (LinFn ∩ ContFn), 𝑇, ( ℋ × {0})) → (∀𝑣 ∈ ℋ (𝑇‘𝑣) = (𝑣 ·ih 𝑤) ↔ ∀𝑣 ∈ ℋ (if(𝑇 ∈ (LinFn ∩ ContFn), 𝑇, ( ℋ × {0}))‘𝑣) = (𝑣 ·ih 𝑤))) |
4 | 3 | reubidv 3393 | . 2 ⊢ (𝑇 = if(𝑇 ∈ (LinFn ∩ ContFn), 𝑇, ( ℋ × {0})) → (∃!𝑤 ∈ ℋ ∀𝑣 ∈ ℋ (𝑇‘𝑣) = (𝑣 ·ih 𝑤) ↔ ∃!𝑤 ∈ ℋ ∀𝑣 ∈ ℋ (if(𝑇 ∈ (LinFn ∩ ContFn), 𝑇, ( ℋ × {0}))‘𝑣) = (𝑣 ·ih 𝑤))) |
5 | inss1 4228 | . . . 4 ⊢ (LinFn ∩ ContFn) ⊆ LinFn | |
6 | 0lnfn 31672 | . . . . . 6 ⊢ ( ℋ × {0}) ∈ LinFn | |
7 | 0cnfn 31667 | . . . . . 6 ⊢ ( ℋ × {0}) ∈ ContFn | |
8 | elin 3964 | . . . . . 6 ⊢ (( ℋ × {0}) ∈ (LinFn ∩ ContFn) ↔ (( ℋ × {0}) ∈ LinFn ∧ ( ℋ × {0}) ∈ ContFn)) | |
9 | 6, 7, 8 | mpbir2an 708 | . . . . 5 ⊢ ( ℋ × {0}) ∈ (LinFn ∩ ContFn) |
10 | 9 | elimel 4597 | . . . 4 ⊢ if(𝑇 ∈ (LinFn ∩ ContFn), 𝑇, ( ℋ × {0})) ∈ (LinFn ∩ ContFn) |
11 | 5, 10 | sselii 3979 | . . 3 ⊢ if(𝑇 ∈ (LinFn ∩ ContFn), 𝑇, ( ℋ × {0})) ∈ LinFn |
12 | inss2 4229 | . . . 4 ⊢ (LinFn ∩ ContFn) ⊆ ContFn | |
13 | 12, 10 | sselii 3979 | . . 3 ⊢ if(𝑇 ∈ (LinFn ∩ ContFn), 𝑇, ( ℋ × {0})) ∈ ContFn |
14 | 11, 13 | riesz4i 31750 | . 2 ⊢ ∃!𝑤 ∈ ℋ ∀𝑣 ∈ ℋ (if(𝑇 ∈ (LinFn ∩ ContFn), 𝑇, ( ℋ × {0}))‘𝑣) = (𝑣 ·ih 𝑤) |
15 | 4, 14 | dedth 4586 | 1 ⊢ (𝑇 ∈ (LinFn ∩ ContFn) → ∃!𝑤 ∈ ℋ ∀𝑣 ∈ ℋ (𝑇‘𝑣) = (𝑣 ·ih 𝑤)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2105 ∀wral 3060 ∃!wreu 3373 ∩ cin 3947 ifcif 4528 {csn 4628 × cxp 5674 ‘cfv 6543 (class class class)co 7412 0cc0 11116 ℋchba 30606 ·ih csp 30609 ContFnccnfn 30640 LinFnclf 30641 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-inf2 9642 ax-cc 10436 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194 ax-addf 11195 ax-mulf 11196 ax-hilex 30686 ax-hfvadd 30687 ax-hvcom 30688 ax-hvass 30689 ax-hv0cl 30690 ax-hvaddid 30691 ax-hfvmul 30692 ax-hvmulid 30693 ax-hvmulass 30694 ax-hvdistr1 30695 ax-hvdistr2 30696 ax-hvmul0 30697 ax-hfi 30766 ax-his1 30769 ax-his2 30770 ax-his3 30771 ax-his4 30772 ax-hcompl 30889 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7674 df-om 7860 df-1st 7979 df-2nd 7980 df-supp 8152 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-2o 8473 df-oadd 8476 df-omul 8477 df-er 8709 df-map 8828 df-pm 8829 df-ixp 8898 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-fsupp 9368 df-fi 9412 df-sup 9443 df-inf 9444 df-oi 9511 df-card 9940 df-acn 9943 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12480 df-z 12566 df-dec 12685 df-uz 12830 df-q 12940 df-rp 12982 df-xneg 13099 df-xadd 13100 df-xmul 13101 df-ioo 13335 df-ico 13337 df-icc 13338 df-fz 13492 df-fzo 13635 df-fl 13764 df-seq 13974 df-exp 14035 df-hash 14298 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-clim 15439 df-rlim 15440 df-sum 15640 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-starv 17219 df-sca 17220 df-vsca 17221 df-ip 17222 df-tset 17223 df-ple 17224 df-ds 17226 df-unif 17227 df-hom 17228 df-cco 17229 df-rest 17375 df-topn 17376 df-0g 17394 df-gsum 17395 df-topgen 17396 df-pt 17397 df-prds 17400 df-xrs 17455 df-qtop 17460 df-imas 17461 df-xps 17463 df-mre 17537 df-mrc 17538 df-acs 17540 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-submnd 18712 df-mulg 18994 df-cntz 19229 df-cmn 19698 df-psmet 21225 df-xmet 21226 df-met 21227 df-bl 21228 df-mopn 21229 df-fbas 21230 df-fg 21231 df-cnfld 21234 df-top 22716 df-topon 22733 df-topsp 22755 df-bases 22769 df-cld 22843 df-ntr 22844 df-cls 22845 df-nei 22922 df-cn 23051 df-cnp 23052 df-lm 23053 df-haus 23139 df-tx 23386 df-hmeo 23579 df-fil 23670 df-fm 23762 df-flim 23763 df-flf 23764 df-xms 24146 df-ms 24147 df-tms 24148 df-cfil 25103 df-cau 25104 df-cmet 25105 df-grpo 30180 df-gid 30181 df-ginv 30182 df-gdiv 30183 df-ablo 30232 df-vc 30246 df-nv 30279 df-va 30282 df-ba 30283 df-sm 30284 df-0v 30285 df-vs 30286 df-nmcv 30287 df-ims 30288 df-dip 30388 df-ssp 30409 df-ph 30500 df-cbn 30550 df-hnorm 30655 df-hba 30656 df-hvsub 30658 df-hlim 30659 df-hcau 30660 df-sh 30894 df-ch 30908 df-oc 30939 df-ch0 30940 df-nlfn 31533 df-cnfn 31534 df-lnfn 31535 |
This theorem is referenced by: riesz2 31753 cnlnadjlem3 31756 cnlnadjlem5 31758 cnvbraval 31797 |
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