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	ubthlem7 8501	Lemma for ubthi 8510. Auxiliary class Q is a vector.
Theorem	ubthlem8 8502	Lemma for ubthi 8510. Compute x in terms of auxiliary vector Q.
Theorem	ubthlem9 8503	Lemma for ubthi 8510. Evaluate the operator value at x in terms of the operator value at Q - p.
Theorem	ubthlem10 8504	Lemma for ubthi 8510. Upper limit for the norm of an operator value at auxiliary vector Q.
Theorem	ubthlem11 8505	Lemma for ubthi 8510. Upper limit for the norm of an operator value at Q - p.
Theorem	ubthlem12 8506	Lemma for ubthi 8510. Upper limit for the norm of an operator value at x.
Theorem	ubthlem13 8507	Lemma for ubthi 8510. Upper bound for the operator norm of any operator T` n.
Theorem	ubthlem14 8508	Lemma for ubthi 8510. The operator norms of the operators T` n have an upper bound.
Theorem	ubthii 8509	Inference from ubthi 8510.
|- 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 8510	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 8511	Lemma for minvecex 8544.
Theorem	minveclem2 8512	Lemma for minvecex 8544.
Theorem	minveclem3 8513	Lemma for minvecex 8544.
Theorem	minveclem4 8514	Lemma for minvecex 8544.
Theorem	minveclem5 8515	Lemma for minvecex 8544.
Theorem	minveclem6 8516	Lemma for minvecex 8544.
Theorem	minveclem7 8517	Lemma for minvecex 8544.
Theorem	minveclem8 8518	Lemma for minvecex 8544.
Theorem	minveclem9 8519	Lemma for minvecex 8544.
Theorem	minveclem10 8520	Lemma for minvecex 8544. The set of reals R is bounded above.
Theorem	minveclem11 8521	Lemma for minvecex 8544.
Theorem	minveclem12 8522	Lemma for minvecex 8544.
Theorem	minveclem13 8523	Lemma for minvecex 8544.
Theorem	minveclem14 8524	Lemma for minvecex 8544.
Theorem	minveclem15 8525	Lemma for minvecex 8544.
Theorem	minveclem16 8526	Lemma for minvecex 8544.
Theorem	minveclem17 8527	Lemma for minvecex 8544.
Theorem	minveclem18 8528	Lemma for minvecex 8544.
Theorem	minveclem19 8529	Lemma for minvecex 8544.
Theorem	minveclem20 8530	Lemma for minvecex 8544.
Theorem	minveclem21 8531	Lemma for minvecex 8544.
Theorem	minveclem22 8532	Lemma for minvecex 8544.
Theorem	minveclem23 8533	Lemma for minvecex 8544. Eliminate H.
Theorem	minveclem24 8534	Lemma for minvecex 8544.
Theorem	minveclem25 8535	Lemma for minvecex 8544.
Theorem	minveclem26 8536	Lemma for minvecex 8544.
Theorem	minveclem27 8537	Lemma for minvecex 8544.
Theorem	minveclem28 8538	Lemma for minvecex 8544.
Theorem	minveclem29 8539	Lemma for minvecex 8544. Sequence f is Cauchy, and since vector subspace W is complete, f therefore converges to a vector in W.
Theorem	minveclem30 8540	Lemma for minvecex 8544.
Theorem	minveclem31 8541	Lemma for minvecex 8544.
Theorem	minveclem32 8542	Lemma for minvecex 8544.
Theorem	minveclem33 8543	Lemma for minvecex 8544.
Theorem	minvecex 8544	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 8545	Lemma for minveceu 8549.
Theorem	minveclem36 8546	Lemma for minveceu 8549.
Theorem	minveclem37 8547	Lemma for minveceu 8549.
Theorem	minveclem38 8548	Lemma for minveceu 8549.
Theorem	minveceu 8549	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 8550	The minimizing vector of minveceu 8549 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 8551	Distance of the minimizing vector of minveceu 8549.
|- 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 8552	The minimizing vector from minveceu 8549 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 8553	Lemma for minvecex2 8554.
Theorem	minvecex2 8554	Existence version of minvecle 8552.
|- 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 8555	Extend class notation with the class of all complex Hilbert spaces.
class CHil
Definition	df-hl 8556	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 8557	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 8558	Every complex Hilbert space is a complex Banach space. (Contributed by Steve Rodriguez, 28-Apr-2007.)
|- (U e. CHil -> U e. CBan)
Theorem	hlph 8559	Every complex Hilbert space is an inner product space (also called a pre-Hilbert space).
|- (U e. CHil -> U e. CPreHil)
Theorem	hlrel 8560	The class of all complex Hilbert spaces is a relation.
|- Rel CHil
Theorem	hlnv 8561	Every complex Hilbert space is a normed complex vector space.
|- (U e. CHil -> U e. NrmCVec)
Theorem	hlnvi 8562	Every complex Hilbert space is a normed complex vector space.
|- U e. CHil   =>   |- U e. NrmCVec
Theorem	hlvc 8563	Every complex Hilbert space is a complex vector space.
|- W = (1st` U)   =>   |- (U e. CHil -> W e. CVec)
Theorem	hlcms 8564	The induced metric on a complex Hilbert space is complete.
|- D = (IndMet` U)   =>   |- (U e. CHil -> D e. CMet)
Theorem	hlmet 8565	The induced metric on a complex Hilbert space.
|- D = (IndMet` U)   =>   |- (U e. CHil -> D e. Met)
Standard axioms for a complex Hilbert space
Theorem	hlex 8566	The base set of a Hilbert space is a set.
|- X = (Base` U)   =>   |- X e. V
Theorem	hladdf 8567	Mapping for Hilbert space vector addition.
|- X = (Base` U)   &   |- G = (+v` U)   =>   |- (U e. CHil -> G:(X X. X)-->X)
Theorem	hlcom 8568	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 8569	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 8570	The Hilbert space zero vector.
|- X = (Base` U)   &   |- Z = (0v` U)   =>   |- (U e. CHil -> Z e. X)
Theorem	hladdid 8571	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 8572	Mapping for Hilbert space scalar multiplication.
|- X = (Base` U)   &   |- S = (.s` U)   =>   |- (U e. CHil -> S:(CC X. X)-->X)
Theorem	hlmulid 8573	Hilbert space scalar multiplication by one.
|- X = (Base` U)   &   |- S = (.s` U)   =>   |- ((U e. CHil /\ A e. X) -> (1SA) = A)
Theorem	hlmulass 8574	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)S