mmtheorems86 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 8501-8600 - Page 86 of 107

Type	Label	Description
Statement
Theorem	minveclem11 8501	Lemma for minvecex 8524.
Theorem	minveclem12 8502	Lemma for minvecex 8524.
Theorem	minveclem13 8503	Lemma for minvecex 8524.
Theorem	minveclem14 8504	Lemma for minvecex 8524.
Theorem	minveclem15 8505	Lemma for minvecex 8524.
Theorem	minveclem16 8506	Lemma for minvecex 8524.
Theorem	minveclem17 8507	Lemma for minvecex 8524.
Theorem	minveclem18 8508	Lemma for minvecex 8524.
Theorem	minveclem19 8509	Lemma for minvecex 8524.
Theorem	minveclem20 8510	Lemma for minvecex 8524.
Theorem	minveclem21 8511	Lemma for minvecex 8524.
Theorem	minveclem22 8512	Lemma for minvecex 8524.
Theorem	minveclem23 8513	Lemma for minvecex 8524. Eliminate H.
Theorem	minveclem24 8514	Lemma for minvecex 8524.
Theorem	minveclem25 8515	Lemma for minvecex 8524.
Theorem	minveclem26 8516	Lemma for minvecex 8524.
Theorem	minveclem27 8517	Lemma for minvecex 8524.
Theorem	minveclem28 8518	Lemma for minvecex 8524.
Theorem	minveclem29 8519	Lemma for minvecex 8524. Sequence f is Cauchy, and since vector subspace W is complete, f therefore converges to a vector in W.
Theorem	minveclem30 8520	Lemma for minvecex 8524.
Theorem	minveclem31 8521	Lemma for minvecex 8524.
Theorem	minveclem32 8522	Lemma for minvecex 8524.
Theorem	minveclem33 8523	Lemma for minvecex 8524.
Theorem	minvecex 8524	Minimizing vector theorem (existence part). There is exactly one vector in a complete subspace W that minimizes the distance to an arbitrary vector A in a parent inner product space. Part of Theorem 3.3-1 of [Kreyszig] p. 144, specialized to subspaces instead of convex subsets. Note that we work with the negative of supremum instead of infimum in order to use theorems we already have available.
|- R = {x | E.y e. Y x = -u(N` (AMy))}   &   |- U e. CPreHil   &   |- M = (-v` U)   &   |- N = (norm` U)   &   |- X = (Base` U)   &   |- W e. (SubSp` U)   &   |- Y = (Base` W)   &   |- A e. X   &   |- P = -usup(R, RR, < )   &   |- (j e. NN -> (F` j) = (N` (AM(f` j))))   &   |- D = (IndMet` W)   &   |- F e. V   &   |- W e. CBan   =>   |- E.a e. Y (N` (AMa)) = P
Theorem	minveclem35 8525	Lemma for minveceu 8529.
Theorem	minveclem36 8526	Lemma for minveceu 8529.
Theorem	minveclem37 8527	Lemma for minveceu 8529.
Theorem	minveclem38 8528	Lemma for minveceu 8529.
Theorem	minveceu 8529	Minimizing vector theorem. There is exactly one vector in a complete subspace W that minimizes the distance to an arbitrary vector A in a parent inner product space. Theorem 3.3-1 of [Kreyszig] p. 144, specialized to subspaces instead of convex subsets. Note that we work with the negative of the supremum of negatives instead of infimum in order to use theorems we already have available.
|- X = (Base` U)   &   |- M = (-v` U)   &   |- N = (norm` U)   &   |- Y = (Base` W)   &   |- R = {x | E.y e. Y x = -u(N` (AMy))}   &   |- P = -usup(R, RR, < )   &   |- U e. CPreHil   &   |- W e. ((SubSp` U) i^i CBan)   &   |- A e. X   =>   |- E!a e. Y (N` (AMa)) = P
Theorem	minveccl 8530	The minimizing vector of minveceu 8529 belongs to the subspace Y.
|- X = (Base` U)   &   |- M = (-v` U)   &   |- N = (norm` U)   &   |- Y = (Base` W)   &   |- R = {x | E.y e. Y x = -u(N` (AMy))}   &   |- P = -usup(R, RR, < )   &   |- U e. CPreHil   &   |- W e. ((SubSp` U) i^i CBan)   &   |- A e. X   &   |- Q = U.{b e. Y | (N` (AMb)) = P}   =>   |- Q e. Y
Theorem	minvecdist 8531	Distance of the minimizing vector of minveceu 8529.
|- X = (Base` U)   &   |- M = (-v` U)   &   |- N = (norm` U)   &   |- Y = (Base` W)   &   |- R = {x | E.y e. Y x = -u(N` (AMy))}   &   |- P = -usup(R, RR, < )   &   |- U e. CPreHil   &   |- W e. ((SubSp` U) i^i CBan)   &   |- A e. X   &   |- Q = U.{b e. Y | (N` (AMb)) = P}   =>   |- (N` (AMQ)) = P
Theorem	minvecle 8532	The minimizing vector from minveceu 8529 has the smallest distance.
|- X = (Base` U)   &   |- M = (-v` U)   &   |- N = (norm` U)   &   |- Y = (Base` W)   &   |- R = {x | E.y e. Y x = -u(N` (AMy))}   &   |- P = -usup(R, RR, < )   &   |- U e. CPreHil   &   |- W e. ((SubSp` U) i^i CBan)   &   |- A e. X   &   |- Q = U.{b e. Y | (N` (AMb)) = P}   =>   |- (B e. Y -> (N` (AMQ)) <_ (N` (AMB)))
Theorem	minveclem39 8533	Lemma for minvecex2 8534.
Theorem	minvecex2 8534	Existence version of minvecle 8532.
|- X = (Base` U)   &   |- M = (-v` U)   &   |- N = (norm` U)   &   |- Y = (Base` W)   &   |- U e. CPreHil   &   |- W e. ((SubSp` U) i^i CBan)   &   |- A e. X   =>   |- E.x e. Y A.y e. Y (N` (AMx)) <_ (N` (AMy))
Complex Hilbert spaces
Definition and basic properties
Syntax	chl 8535	Extend class notation with the class of all complex Hilbert spaces.
class CHil
Definition	df-hl 8536	Define the class of all complex Hilbert spaces. A Hilbert space is a Banach space which is also an inner product space.
|- CHil = (CBan i^i CPreHil)
Theorem	ishl 8537	The predicate "is a complex Hilbert space." A Hilbert space is a Banach space which is also an inner product space, i.e. whose norm satisfies the parallelogram law. (Contributed by Steve Rodriguez, 28-Apr-2007.)
|- (U e. CHil <-> (U e. CBan /\ U e. CPreHil))
Theorem	hlbn 8538	Every complex Hilbert space is a complex Banach space. (Contributed by Steve Rodriguez, 28-Apr-2007.)
|- (U e. CHil -> U e. CBan)
Theorem	hlph 8539	Every complex Hilbert space is an inner product space (also called a pre-Hilbert space).
|- (U e. CHil -> U e. CPreHil)
Theorem	hlrel 8540	The class of all complex Hilbert spaces is a relation.
|- Rel CHil
Theorem	hlnv 8541	Every complex Hilbert space is a normed complex vector space.
|- (U e. CHil -> U e. NrmCVec)
Theorem	hlnvi 8542	Every complex Hilbert space is a normed complex vector space.
|- U e. CHil   =>   |- U e. NrmCVec
Theorem	hlvc 8543	Every complex Hilbert space is a complex vector space.
|- W = (1st` U)   =>   |- (U e. CHil -> W e. CVec)
Theorem	hlcms 8544	The induced metric on a complex Hilbert space is complete.
|- D = (IndMet` U)   =>   |- (U e. CHil -> D e. CMet)
Standard axioms for a complex Hilbert space
Theorem	hlex 8545	The base set of a Hilbert space is a set.
|- X = (Base` U)   =>   |- X e. V
Theorem	hladdf 8546	Mapping for Hilbert space vector addition.
|- X = (Base` U)   &   |- G = (+v` U)   =>   |- (U e. CHil -> G:(X X. X)-->X)
Theorem	hlcom 8547	Hilbert space vector addition is commutative.
|- X = (Base` U)   &   |- G = (+v` U)   =>   |- ((U e. CHil /\ A e. X /\ B e. X) -> (AGB) = (BGA))
Theorem	hlass 8548	Hilbert space vector addition is associative.
|- X = (Base` U)   &   |- G = (+v` U)   =>   |- ((U e. CHil /\ (A e. X /\ B e. X /\ C e. X)) -> ((AGB)GC) = (AG(BGC)))
Theorem	hl0cl 8549	The Hilbert space zero vector.
|- X = (Base` U)   &   |- Z = (0v` U)   =>   |- (U e. CHil -> Z e. X)
Theorem	hladdid 8550	Hilbert space addition with the zero vector.
|- X = (Base` U)   &   |- G = (+v` U)   &   |- Z = (0v` U)   =>   |- ((U e. CHil /\ A e. X) -> (AGZ) = A)
Theorem	hlmulf 8551	Mapping for Hilbert space scalar multiplication.
|- X = (Base` U)   &   |- S = (.s` U)   =>   |- (U e. CHil -> S:(CC X. X)-->X)
Theorem	hlmulid 8552	Hilbert space scalar multiplication by one.
|- X = (Base` U)   &   |- S = (.s` U)   =>   |- ((U e. CHil /\ A e. X) -> (1SA) = A)
Theorem	hlmulass 8553	Hilbert space scalar multiplication associative law.
|- X = (Base` U)   &   |- S = (.s` U)   =>   |- ((U e. CHil /\ (A e. CC /\ B e. CC /\ C e. X)) -> ((A x. B)SC) = (AS(BSC)))
Theorem	hldi 8554	Hilbert space scalar multiplication distributive law.
|- X = (Base` U)   &   |- G = (+v` U)   &   |- S = (.s` U)   =>   |- ((U e. CHil /\ (A e. CC /\ B e. X /\ C e. X)) -> (AS(BGC)) = ((ASB)G(ASC)))
Theorem	hldir 8555	Hilbert space scalar multiplication distributive law.
|- X = (Base` U)   &   |- G = (+v` U)   &   |- S = (.s` U)   =>   |- ((U e. CHil /\ (A e. CC /\ B e. CC /\ C e. X)) -> ((A + B)SC) = ((ASC)G(BSC)))
Theorem	hlmul0 8556	Hilbert space scalar multiplication by zero.
|- X = (Base` U)   &   |- S = (.s` U)   &   |- Z = (0v` U)   =>   |- ((U e. CHil /\ A e. X) -> (0SA) = Z)
Theorem	hlipf 8557	Mapping for Hilbert space inner product.
|- X = (Base` U)   &   |- P = (.i` U)   =>   |- (U e. CHil -> P:(X X. X)-->CC)
Theorem	hlipcj 8558	Conjugate law for Hilbert space inner product.
|- X = (Base` U)   &   |- P = (.i` U)   =>   |- ((U e. CHil /\ A e. X /\ B e. X) -> (APB) = (*` (BPA)))
Theorem	hlipdir 8559	Distributive law for Hilbert space inner product.
|- X = (Base` U)   &   |- G = (+v` U)   &   |- P = (.i` U)   =>   |- ((U e. CHil /\ (A e. X /\ B e. X /\ C e. X)) -> ((AGB)PC) = ((APC) + (BPC)))