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	minveclem3 8501	Lemma for minvecex 8532.
Theorem	minveclem4 8502	Lemma for minvecex 8532.
Theorem	minveclem5 8503	Lemma for minvecex 8532.
Theorem	minveclem6 8504	Lemma for minvecex 8532.
Theorem	minveclem7 8505	Lemma for minvecex 8532.
Theorem	minveclem8 8506	Lemma for minvecex 8532.
Theorem	minveclem9 8507	Lemma for minvecex 8532.
Theorem	minveclem10 8508	Lemma for minvecex 8532. The set of reals R is bounded above.
Theorem	minveclem11 8509	Lemma for minvecex 8532.
Theorem	minveclem12 8510	Lemma for minvecex 8532.
Theorem	minveclem13 8511	Lemma for minvecex 8532.
Theorem	minveclem14 8512	Lemma for minvecex 8532.
Theorem	minveclem15 8513	Lemma for minvecex 8532.
Theorem	minveclem16 8514	Lemma for minvecex 8532.
Theorem	minveclem17 8515	Lemma for minvecex 8532.
Theorem	minveclem18 8516	Lemma for minvecex 8532.
Theorem	minveclem19 8517	Lemma for minvecex 8532.
Theorem	minveclem20 8518	Lemma for minvecex 8532.
Theorem	minveclem21 8519	Lemma for minvecex 8532.
Theorem	minveclem22 8520	Lemma for minvecex 8532.
Theorem	minveclem23 8521	Lemma for minvecex 8532. Eliminate H.
Theorem	minveclem24 8522	Lemma for minvecex 8532.
Theorem	minveclem25 8523	Lemma for minvecex 8532.
Theorem	minveclem26 8524	Lemma for minvecex 8532.
Theorem	minveclem27 8525	Lemma for minvecex 8532.
Theorem	minveclem28 8526	Lemma for minvecex 8532.
Theorem	minveclem29 8527	Lemma for minvecex 8532. Sequence f is Cauchy, and since vector subspace W is complete, f therefore converges to a vector in W.
Theorem	minveclem30 8528	Lemma for minvecex 8532.
Theorem	minveclem31 8529	Lemma for minvecex 8532.
Theorem	minveclem32 8530	Lemma for minvecex 8532.
Theorem	minveclem33 8531	Lemma for minvecex 8532.
Theorem	minvecex 8532	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 8533	Lemma for minveceu 8537.
Theorem	minveclem36 8534	Lemma for minveceu 8537.
Theorem	minveclem37 8535	Lemma for minveceu 8537.
Theorem	minveclem38 8536	Lemma for minveceu 8537.
Theorem	minveceu 8537	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 8538	The minimizing vector of minveceu 8537 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 8539	Distance of the minimizing vector of minveceu 8537.
|- 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 8540	The minimizing vector from minveceu 8537 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 8541	Lemma for minvecex2 8542.
Theorem	minvecex2 8542	Existence version of minvecle 8540.
|- 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 8543	Extend class notation with the class of all complex Hilbert spaces.
class CHil
Definition	df-hl 8544	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 8545	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 8546	Every complex Hilbert space is a complex Banach space. (Contributed by Steve Rodriguez, 28-Apr-2007.)
|- (U e. CHil -> U e. CBan)
Theorem	hlph 8547	Every complex Hilbert space is an inner product space (also called a pre-Hilbert space).
|- (U e. CHil -> U e. CPreHil)
Theorem	hlrel 8548	The class of all complex Hilbert spaces is a relation.
|- Rel CHil
Theorem	hlnv 8549	Every complex Hilbert space is a normed complex vector space.
|- (U e. CHil -> U e. NrmCVec)
Theorem	hlnvi 8550	Every complex Hilbert space is a normed complex vector space.
|- U e. CHil   =>   |- U e. NrmCVec
Theorem	hlvc 8551	Every complex Hilbert space is a complex vector space.
|- W = (1st` U)   =>   |- (U e. CHil -> W e. CVec)
Theorem	hlcms 8552	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 8553	The base set of a Hilbert space is a set.
|- X = (Base` U)   =>   |- X e. V
Theorem	hladdf 8554	Mapping for Hilbert space vector addition.
|- X = (Base` U)   &   |- G = (+v` U)   =>   |- (U e. CHil -> G:(X X. X)-->X)
Theorem	hlcom 8555	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 8556	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 8557	The Hilbert space zero vector.
|- X = (Base` U)   &   |- Z = (0v` U)   =>   |- (U e. CHil -> Z e. X)
Theorem	hladdid 8558	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 8559	Mapping for Hilbert space scalar multiplication.
|- X = (Base` U)   &   |- S = (.s` U)   =>   |- (U e. CHil -> S:(CC X. X)-->X)
Theorem	hlmulid 8560	Hilbert space scalar multiplication by one.
|- X = (Base` U)   &   |- S = (.s` U)   =>   |- ((U e. CHil /\ A e. X) -> (1SA) = A)
Theorem	hlmulass 8561	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 8562	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 8563	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 8564	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 8565	Mapping for Hilbert space inner product.
|- X = (Base` U)   &   |- P = (.i` U)   =>   |- (U e. CHil -> P:(X X. X)-->CC)
Theorem	hlipcj 8566	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 8567	Distributive law for Hilbert space inner product.
|- X = (Base` U)   &   |- G = (+v` U)   &   |- P