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	ubthlem14 8501	Lemma for ubthi 8503. The operator norms of the operators T` n have an upper bound.
Theorem	ubthii 8502	Inference from ubthi 8503.
|- X = (Base` U)   &   |- M = (norm` W)   &   |- N = (UnormOpW)   &   |- B = (U BLnOp W)   &   |- U e. CBan   &   |- W e. NrmCVec   &   |- T:NN-->B   =>   |- (A.x e. X E.c e. RR A.n e. NN (M` ((T` n)` x)) <_ c -> E.d e. RR A.n e. NN (N` (T` n)) <_ d)
Theorem	ubthi 8503	Uniform Boundedness Theorem. Let T be a sequence of bounded linear operators on a Banach space. If, for every vector x, the norms of the operators' values are bounded, then the operators' norms are also bounded. Theorem 4.7-3 of [Kreyszig] p. 249. See also http://en.wikipedia.org/wiki/Uniform_boundedness_principle.
|- X = (Base` U)   &   |- M = (norm` W)   &   |- N = (UnormOpW)   &   |- B = (U BLnOp W)   &   |- U e. CBan   &   |- W e. NrmCVec   =>   |- ((T:NN-->B /\ A.x e. X E.c e. RR A.n e. NN (M` ((T` n)` x)) <_ c) -> E.d e. RR A.n e. NN (N` (T` n)) <_ d)
Minimizing Vector Theorem
Theorem	minveclem1 8504	Lemma for minvecex 8537.
Theorem	minveclem2 8505	Lemma for minvecex 8537.
Theorem	minveclem3 8506	Lemma for minvecex 8537.
Theorem	minveclem4 8507	Lemma for minvecex 8537.
Theorem	minveclem5 8508	Lemma for minvecex 8537.
Theorem	minveclem6 8509	Lemma for minvecex 8537.
Theorem	minveclem7 8510	Lemma for minvecex 8537.
Theorem	minveclem8 8511	Lemma for minvecex 8537.
Theorem	minveclem9 8512	Lemma for minvecex 8537.
Theorem	minveclem10 8513	Lemma for minvecex 8537. The set of reals R is bounded above.
Theorem	minveclem11 8514	Lemma for minvecex 8537.
Theorem	minveclem12 8515	Lemma for minvecex 8537.
Theorem	minveclem13 8516	Lemma for minvecex 8537.
Theorem	minveclem14 8517	Lemma for minvecex 8537.
Theorem	minveclem15 8518	Lemma for minvecex 8537.
Theorem	minveclem16 8519	Lemma for minvecex 8537.
Theorem	minveclem17 8520	Lemma for minvecex 8537.
Theorem	minveclem18 8521	Lemma for minvecex 8537.
Theorem	minveclem19 8522	Lemma for minvecex 8537.
Theorem	minveclem20 8523	Lemma for minvecex 8537.
Theorem	minveclem21 8524	Lemma for minvecex 8537.
Theorem	minveclem22 8525	Lemma for minvecex 8537.
Theorem	minveclem23 8526	Lemma for minvecex 8537. Eliminate H.
Theorem	minveclem24 8527	Lemma for minvecex 8537.
Theorem	minveclem25 8528	Lemma for minvecex 8537.
Theorem	minveclem26 8529	Lemma for minvecex 8537.
Theorem	minveclem27 8530	Lemma for minvecex 8537.
Theorem	minveclem28 8531	Lemma for minvecex 8537.
Theorem	minveclem29 8532	Lemma for minvecex 8537. Sequence f is Cauchy, and since vector subspace W is complete, f therefore converges to a vector in W.
Theorem	minveclem30 8533	Lemma for minvecex 8537.
Theorem	minveclem31 8534	Lemma for minvecex 8537.
Theorem	minveclem32 8535	Lemma for minvecex 8537.
Theorem	minveclem33 8536	Lemma for minvecex 8537.
Theorem	minvecex 8537	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 8538	Lemma for minveceu 8542.
Theorem	minveclem36 8539	Lemma for minveceu 8542.
Theorem	minveclem37 8540	Lemma for minveceu 8542.
Theorem	minveclem38 8541	Lemma for minveceu 8542.
Theorem	minveceu 8542	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 8543	The minimizing vector of minveceu 8542 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 8544	Distance of the minimizing vector of minveceu 8542.
|- 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 8545	The minimizing vector from minveceu 8542 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 8546	Lemma for minvecex2 8547.
Theorem	minvecex2 8547	Existence version of minvecle 8545.
|- 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 8548	Extend class notation with the class of all complex Hilbert spaces.
class CHil
Definition	df-hl 8549	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 8550	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 8551	Every complex Hilbert space is a complex Banach space. (Contributed by Steve Rodriguez, 28-Apr-2007.)
|- (U e. CHil -> U e. CBan)
Theorem	hlph 8552	Every complex Hilbert space is an inner product space (also called a pre-Hilbert space).
|- (U e. CHil -> U e. CPreHil)
Theorem	hlrel 8553	The class of all complex Hilbert spaces is a relation.
|- Rel CHil
Theorem	hlnv 8554	Every complex Hilbert space is a normed complex vector space.
|- (U e. CHil -> U e. NrmCVec)
Theorem	hlnvi 8555	Every complex Hilbert space is a normed complex vector space.
|- U e. CHil   =>   |- U e. NrmCVec
Theorem	hlvc 8556	Every complex Hilbert space is a complex vector space.
|- W = (1st` U)   =>   |- (U e. CHil -> W e. CVec)
Theorem	hlcms 8557	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 8558	The base set of a Hilbert space is a set.
|- X = (Base` U)   =>   |- X e. V
Theorem	hladdf 8559	Mapping for Hilbert space vector addition.
|- X = (Base` U)   &   |- G = (+v` U)   =>   |- (U e. CHil -> G:(X X. X)-->X)
Theorem	hlcom 8560	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 8561	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 8562	The Hilbert space zero vector.
|- X = (Base` U)   &   |- Z = (0v` U)   =>   |- (U e. CHil -> Z e. X)
Theorem	hladdid 8563	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 8564	Mapping for Hilbert space scalar multiplication.
|- X = (Base` U)   &   |- S = (.s` U)   =>   |- (U e. CHil -> S:(CC X. X)-->X)
Theorem	hlmulid 8565	Hilbert space scalar multiplication by one.
|- X = (Base` U)   &   |- S = (.s` U)   =>   |- ((U e. CHil /\ A e. X) -> (1SA) = A)
Theorem	hlmulass 8566	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 8567	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)