mmtheorems91 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 9001-9100 - Page 91 of 107

Type	Label	Description
Statement
Theorem	hilhh 9001	Deduce the structure of Hilbert space from its components.
|- H~ = (Base` U)   &   |- +h = (+v` U)   &   |- .h = (.s` U)   &   |- .ih = (.i` U)   &   |- U e. NrmCVec   =>   |- U = <.<. +h , .h >., normh>.
Theorem	hhnv 9002	Hilbert space is a normed complex vector space.
|- U = <.<. +h , .h >., normh>.   =>   |- U e. NrmCVec
Theorem	hhva 9003	The group (addition) operation of Hilbert space.
|- U = <.<. +h , .h >., normh>.   =>   |- +h = (+v` U)
Theorem	hhba 9004	The base set of Hilbert space.
|- U = <.<. +h , .h >., normh>.   =>   |- H~ = (Base` U)
Theorem	hh0v 9005	The zero vector of Hilbert space.
|- U = <.<. +h , .h >., normh>.   =>   |- 0h = (0v` U)
Theorem	hhsm 9006	The scalar product operation of Hilbert space.
|- U = <.<. +h , .h >., normh>.   =>   |- .h = (.s` U)
Theorem	hhvs 9007	The vector subtraction operation of Hilbert space.
|- U = <.<. +h , .h >., normh>.   =>   |- -h = (-v` U)
Theorem	hhnm 9008	The norm function of Hilbert space.
|- U = <.<. +h , .h >., normh>.   =>   |- normh = (norm` U)
Theorem	hhims 9009	The induced metric of Hilbert space.
|- U = <.<. +h , .h >., normh>.   &   |- D = (normh o. -h )   =>   |- D = (IndMet` U)
Theorem	hhims2 9010	Hilbert space distance metric.
|- U = <.<. +h , .h >., normh>.   &   |- D = (IndMet` U)   =>   |- D = (normh o. -h )
Theorem	hhmet 9011	The induced metric of Hilbert space.
|- U = <.<. +h , .h >., normh>.   &   |- D = (IndMet` U)   =>   |- D e. Met
Theorem	hhmetba 9012	The base set for the metric for Hilbert space.
|- U = <.<. +h , .h >., normh>.   &   |- D = (IndMet` U)   =>   |- H~ = dom dom D
Theorem	hhmetdval 9013	Value of the distance function of the metric space of Hilbert space.
|- U = <.<. +h , .h >., normh>.   &   |- D = (IndMet` U)   =>   |- ((A e. H~ /\ B e. H~) -> (ADB) = (normh` (A -h B)))
Theorem	hhip 9014	The inner product operation of Hilbert space.
|- U = <.<. +h , .h >., normh>.   =>   |- .ih = (.i` U)
Theorem	hhph 9015	The Hilbert space of the Hilbert Space Explorer is an inner product space.
|- U = <.<. +h , .h >., normh>.   =>   |- U e. CPreHil
Bunjakovaskij-Cauchy-Schwarz inequality
Theorem	bcsALT 9016	Bunjakovaskij-Cauchy-Schwarz inequality. Remark 3.4 of [Beran] p. 98.
|- A e. H~   &   |- B e. H~   =>   |- (abs` (A .ih B)) <_ ((normh` A) x. (normh` B))
Theorem	bcsHIL 9017	Bunjakovaskij-Cauchy-Schwarz inequality. Remark 3.4 of [Beran] p. 98. (Proved from ZFC version.)
|- A e. H~   &   |- B e. H~   =>   |- (abs` (A .ih B)) <_ ((normh` A) x. (normh` B))
Theorem	bcst 9018	Bunjakovaskij-Cauchy-Schwarz inequality. Remark 3.4 of [Beran] p. 98.
|- ((A e. H~ /\ B e. H~) -> (abs` (A .ih B)) <_ ((normh` A) x. (normh` B)))
Theorem	bcs2t 9019	Corollary of the Bunjakovaskij-Cauchy-Schwarz inequality bcsHIL 9017.
|- ((A e. H~ /\ B e. H~ /\ (normh` A) <_ 1) -> (abs` (A .ih B)) <_ (normh` B))
Theorem	bcs3t 9020	Corollary of the Bunjakovaskij-Cauchy-Schwarz inequality bcsHIL 9017.
|- ((A e. H~ /\ B e. H~ /\ (normh` B) <_ 1) -> (abs` (A .ih B)) <_ (normh` A))
Cauchy sequences and limits
Theorem	hcau 9021	Member of the set of Cauchy sequences on a Hilbert space. Definition for Cauchy sequence in [Beran] p. 96.
|- (F e. Cauchy <-> (F:NN-->H~ /\ A.x e. RR (0 < x -> E.y e. NN A.z e. NN A.w e. NN ((y <_ z /\ y <_ w) -> (normh` ((F` z) -h (F` w))) < x))))
Theorem	hcauseq 9022	A Cauchy sequences on a Hilbert space is a sequence.
|- (F e. Cauchy -> F:NN-->H~)
Theorem	hcaucvg 9023	A Cauchy sequence on a Hilbert space converges.
|- (F e. Cauchy -> A.x e. RR (0 < x -> E.y e. NN A.z e. NN A.w e. NN ((y <_ z /\ y <_ w) -> (normh` ((F` z) -h (F` w))) < x)))
Theorem	seq1hcau 9024	A sequence on a Hilbert space is a Cauchy sequence if it converges.
|- (F:NN-->H~ -> (F e. Cauchy <-> A.x e. RR (0 < x -> E.y e. NN A.z e. NN A.w e. NN ((y <_ z /\ y <_ w) -> (normh` ((F` z) -h (F` w))) < x))))
Theorem	hcau2 9025	Alternate representation of a Hilbert space Cauchy sequence.
|- (F e. Cauchy <-> (F:NN-->H~ /\ A.x e. RR (0 < x -> E.j e. NN A.k e. NN (j <_ k -> (normh` ((F` k) -h (F` j))) < x))))
Theorem	hlim 9026	Express the predicate: The limit of vector sequence F in a Hilbert space is A, i.e. F converges to A. This means that for any real x, no matter how small, there always exists an integer y such that the norm of any later vector in the sequence minus the limit is less than x. Definition of converge in [Beran] p. 96.
|- F e. V   &   |- A e. V   =>   |- (F ~~>v A <-> ((F:NN-->H~ /\ A e. H~) /\ A.x e. RR (0 < x -> E.y e. NN A.z e. NN (y <_ z -> (normh` ((F` z) -h A)) < x))))
Theorem	hlimseq 9027	A sequence with a limit on a Hilbert space is a sequence.
|- F e. V   &   |- A e. V   =>   |- (F ~~>v A -> F:NN-->H~)
Theorem	hlimvec 9028	Closure of the limit of a sequence on Hilbert space.
|- F e. V   &   |- A e. V   =>   |- (F ~~>v A -> A e. H~)
Theorem	hlimconv 9029	Convergence of a sequence on a Hilbert space.
|- F e. V   &   |- A e. V   =>   |- (F ~~>v A -> A.x e. RR (0 < x -> E.y e. NN A.z e. NN (y <_ z -> (normh` ((F` z) -h A)) < x)))
Theorem	hlim2 9030	The limit of a sequence on a Hilbert space.
|- ((F:NN-->H~ /\ A e. H~) -> (F ~~>v A <-> A.x e. RR (0 < x -> E.y e. NN A.z e. NN (y <_ z -> (normh` ((F` z) -h A)) < x))))
Derivation of the completeness axiom from ZF set theory
Theorem	hilmet 9031	The Hilbert space norm determines a metric space.
|- D = (normh o. -h )   =>   |- D e. Met
Theorem	hilmetdval 9032	Value of the distance function of the metric space of Hilbert space.
|- D = (normh o. -h )   =>   |- ((A e. H~ /\ B e. H~) -> (ADB) = (normh` (A -h B)))
Theorem	hilmetba 9033	The base set for the metric for Hilbert space.
|- D = (normh o. -h )   =>   |- H~ = dom dom D
Theorem	hilims 9034	Hilbert space distance metric.
|- H~ = (Base` U)   &   |- +h = (+v` U)   &   |- .h = (.s` U)   &   |- .ih = (.i` U)   &   |- D = (IndMet` U)   &   |- U e. NrmCVec   =>   |- D = (normh o. -h )
Theorem	hillim 9035	Hilbert space limit in terms of metric space limit.
|- H~ = (Base` U)   &   |- +h = (+v` U)   &   |- .h = (.s` U)   &   |- .ih = (.i` U)   &   |- D = (IndMet` U)   &   |- U e. NrmCVec   =>   |- ~~>v = ((~~>m` D) |` (H~ ^m NN))
Theorem	hilcau 9036	Hilbert space Cauchy sequences in terms of metric space Cauchy sequences.
|- H~ = (Base` U)   &   |- +h = (+v` U)   &   |- .h = (.s` U)   &   |- .ih = (.i` U)   &   |- D = (IndMet` U)   &   |- U e. NrmCVec   =>   |- Cauchy = ((Cau` D) i^i (H~ ^m NN))
Theorem	hhlm 9037	The limit sequences of Hilbert space.
|- U = <.<. +h , .h >., normh>.   &   |- D = (IndMet` U)   =>   |- ~~>v = ((~~>m` D) |` (H~ ^m NN))
Theorem	hhcau 9038	The Cauchy sequences of Hilbert space.
|- U = <.<. +h , .h >., normh>.   &   |- D = (IndMet` U)   =>   |- Cauchy = ((Cau` D) i^i (H~ ^m NN))
Theorem	hhcmpl 9039	Lemma used for derivation of the completeness axiom ax-hcompl 9041 from ZFC Hilbert space theory.
|- U = <.<. +h , .h >., normh>.   &   |- D = (IndMet` U)   &   |- (F e. (Cau` D) -> E.x e. H~ F(~~>m` D)x)   =>   |- (F e. Cauchy -> E.x e. H~ F ~~>v x)
Theorem	hilcompl 9040	Lemma used for derivation of the completeness axiom ax-hcompl 9041 from ZFC Hilbert space theory. The first 5 hypotheses would be satisfied by the definitions described in ax-hilex 8839; the 6th would be satisfied by eqid 1473; the 7th by a given fixed Hilbert space; and the last by theorem hlcompl 8572.
|- H~ = (Base` U)   &   |- +h = (+v` U)   &   |- .h = (.s` U)   &   |- .ih = (.i` U)   &   |- D = (IndMet` U)   &   |- U e. CHil   &   |- (F e. (Cau` D) -> E.x e. H~ F(~~>m` D)x)   =>   |- (F e. Cauchy -> E.x e. H~ F ~~>v x)
Completeness postulate for a Hilbert space
Axiom	ax-hcompl 9041	Completeness of a Hilbert space.
|- (F e. Cauchy -> E.x e. H~ F ~~>v x)
Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces
Theorem	hhcms 9042	The Hilbert space induced metric determines a complete metric space.
|- U = <.<. +h , .h >., normh>.   &   |- D = (IndMet` U)   =>   |- D e. CMet
Theorem	hhhl 9043	The Hilbert space structure is a complex Hilbert space.
|- U = <.<. +h ,