mmtheorems89 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 8801-8900 - Page 89 of 123

Type	Label	Description
Statement
Theorem	ubthlem13OLD 8801	Lemma for ubthi 8804. Upper bound for the operator norm of any operator T` n.
Theorem	ubthlem14 8802	Lemma for ubthi 8804. The operator norms of the operators T` n have an upper bound.
Theorem	ubthii 8803	Inference from ubthi 8804.
|- X = (BaseSet` 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 8804	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 = (BaseSet` 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 8805	Lemma for minvecex 8838.
Theorem	minveclem2 8806	Lemma for minvecex 8838.
Theorem	minveclem3 8807	Lemma for minvecex 8838.
Theorem	minveclem4 8808	Lemma for minvecex 8838.
Theorem	minveclem5 8809	Lemma for minvecex 8838.
Theorem	minveclem6 8810	Lemma for minvecex 8838.
Theorem	minveclem7 8811	Lemma for minvecex 8838.
Theorem	minveclem8 8812	Lemma for minvecex 8838.
Theorem	minveclem9 8813	Lemma for minvecex 8838.
Theorem	minveclem10 8814	Lemma for minvecex 8838. The set of reals R is bounded above.
Theorem	minveclem11 8815	Lemma for minvecex 8838.
Theorem	minveclem12 8816	Lemma for minvecex 8838.
Theorem	minveclem13 8817	Lemma for minvecex 8838.
Theorem	minveclem14 8818	Lemma for minvecex 8838.
Theorem	minveclem15 8819	Lemma for minvecex 8838.
Theorem	minveclem16 8820	Lemma for minvecex 8838.
Theorem	minveclem17 8821	Lemma for minvecex 8838.
Theorem	minveclem18 8822	Lemma for minvecex 8838.
Theorem	minveclem19 8823	Lemma for minvecex 8838.
Theorem	minveclem20 8824	Lemma for minvecex 8838.
Theorem	minveclem21 8825	Lemma for minvecex 8838.
Theorem	minveclem22 8826	Lemma for minvecex 8838.
Theorem	minveclem23 8827	Lemma for minvecex 8838. Eliminate H.
Theorem	minveclem24 8828	Lemma for minvecex 8838.
Theorem	minveclem25 8829	Lemma for minvecex 8838.
Theorem	minveclem26 8830	Lemma for minvecex 8838.
Theorem	minveclem27 8831	Lemma for minvecex 8838.
Theorem	minveclem28 8832	Lemma for minvecex 8838.
Theorem	minveclem29 8833	Lemma for minvecex 8838. Sequence f is Cauchy, and since vector subspace W is complete, f therefore converges to a vector in W.
Theorem	minveclem30 8834	Lemma for minvecex 8838.
Theorem	minveclem31 8835	Lemma for minvecex 8838.
Theorem	minveclem32 8836	Lemma for minvecex 8838.
Theorem	minveclem33 8837	Lemma for minvecex 8838.
Theorem	minvecex 8838	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 = (BaseSet` U)   &   |- W e. (SubSp` U)   &   |- Y = (BaseSet` 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 8839	Lemma for minveceu 8843.
Theorem	minveclem36 8840	Lemma for minveceu 8843.
Theorem	minveclem37 8841	Lemma for minveceu 8843.
Theorem	minveclem38 8842	Lemma for minveceu 8843.
Theorem	minveceu 8843	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 = (BaseSet` U)   &   |- M = (-v` U)   &   |- N = (norm` U)   &   |- Y = (BaseSet` 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 8844	The minimizing vector of minveceu 8843 belongs to the subspace Y.
|- X = (BaseSet` U)   &   |- M = (-v` U)   &   |- N = (norm` U)   &   |- Y = (BaseSet` 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 8845	Distance of the minimizing vector of minveceu 8843.
|- X = (BaseSet` U)   &   |- M = (-v` U)   &   |- N = (norm` U)   &   |- Y = (BaseSet` 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 8846	The minimizing vector from minveceu 8843 has the smallest distance.
|- X = (BaseSet` U)   &   |- M = (-v` U)   &   |- N = (norm` U)   &   |- Y = (BaseSet` 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 8847	Lemma for minvecex2 8848.
Theorem	minvecex2 8848	Existence version of minvecle 8846.
|- X = (BaseSet` U)   &   |- M = (-v` U)   &   |- N = (norm` U)   &   |- Y = (BaseSet` 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 8849	Extend class notation with the class of all complex Hilbert spaces.
class CHil
Definition	df-hl 8850	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 8851	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 8852	Every complex Hilbert space is a complex Banach space. (Contributed by Steve Rodriguez, 28-Apr-2007.)
|- (U e. CHil -> U e. CBan)
Theorem	hlph 8853	Every complex Hilbert space is an inner product space (also called a pre-Hilbert space).
|- (U e. CHil -> U e. CPreHil)
Theorem	hlrel 8854	The class of all complex Hilbert spaces is a relation.
|- Rel CHil
Theorem	hlnv 8855	Every complex Hilbert space is a normed complex vector space.
|- (U e. CHil -> U e. NrmCVec)
Theorem	hlnvi 8856	Every complex Hilbert space is a normed complex vector space.
|- U e. CHil   =>   |- U e. NrmCVec
Theorem	hlvc 8857	Every complex Hilbert space is a complex vector space.
|- W = (1st` U)   =>   |- (U e. CHil -> W e. CVec)
Theorem	hlcms 8858	The induced metric on a complex Hilbert space is complete.
|- D = (IndMet` U)   =>   |- (U e. CHil -> D e. CMet)
Theorem	hlmet 8859	The induced metric on a complex Hilbert space.
|- D = (IndMet` U)   =>   |- (U e. CHil -> D e. Met)
Theorem	hlpar2 8860	The parallelogram law satified by Hilbert space vectors. (Contributed by Steve Rodriguez, 28-Apr-2007.)
|- X = (BaseSet` U)   &   |- G = (+v` U)   &   |- M = (-v` U)   &   |- N = (norm` U)   =>   |- ((U e. CHil /\ A e. X /\ B e. X) -> (((N` (AGB))^2) + ((N` (AMB))^2)) = (2 x. (((N` A)^2) + ((N` B)^2))))
Theorem	hlpar 8861	The parallelogram law satified by Hilbert space vectors. (Contributed by Steve Rodriguez, 28-Apr-2007.)
|- X = (BaseSet` U)   &   |- G = (+v` U)   &   |- S = (.s` U)   &   |- N = (norm` U)   =>   |- ((U e. CHil /\ A e. X /\ B e. X) -> (((N` (AGB))^2) + ((N` (AG(-u1SB)))^2)) = (2 x. (((N` A)^2) + ((N` B)^2))))
Standard axioms for a complex Hilbert space
Theorem	hlex 8862	The base set of a Hilbert space is a set.
|- X = (BaseSet` U)   =>   |- X e. V
Theorem	hladdf 8863	Mapping for Hilbert space vector addition.
|- X = (BaseSet` U)   &   |- G = (+v` U)   =>   |- (U e. CHil -> G:(X X. X)-->X)
Theorem	hlcom 8864	Hilbert space vector addition is commutative.
|- X = (BaseSet` U)   &   |- G = (+v` U)   =>   |- ((U e. CHil /\ A e. X /\ B e. X) -> (AGB) = (BGA))
Theorem	hlass 8865	Hilbert space vector addition is associative.
|- X = (BaseSet` U)   &   |- G = (+v` U)   =>   |- ((U e. CHil /\ (A e. X /\ B e. X /\ C e. X)) -> ((AGB)GC) = (AG(BGC)))
Theorem	hl0cl 8866	The Hilbert space zero vector.
|- X = (BaseSet` U)   &   |- Z = (0v` U)   =>   |- (U e. CHil -> Z e. X)
Theorem	hladdid 8867	Hilbert space addition with the zero vector.
|- X = (BaseSet` U)   &   |- G = (+v` U)   &   |- Z = (0v` U)   =>   |- ((U e. CHil /\ A e. X) -> (AGZ) = A)
Theorem	hlmulf 8868	Mapping for Hilbert space scalar multiplication.
|- X = (BaseSet` U)   &   |- S = (.s` U)   =>   |- (U e. CHil -> S:(CC X. X)-->X)
Theorem	hlmulid 8869	Hilbert space scalar multiplication by one.
|- X = (BaseSet` U)   &   |- S = (.s` U)   =>   |- ((U e. CHil /\ A e. X) -> (1SA) = A)
Theorem	hlmulass 8870	Hilbert space scalar multiplication associative law.
|- X = (BaseSet` U)   &   |- S = (.s` U)   =>   |- ((U e. CHil /\ (A e. CC /\ B e. CC /\ C e. X)) -> ((A x. B)SC) = (AS(BSC)))
Theorem	hldi 8871	Hilbert space scalar multiplication distributive law.
|- X = (BaseSet` 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 8872	Hilbert space scalar multiplication distributive law.
|- X = (BaseSet` 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 8873	Hilbert space scalar multiplication by zero.
|- X = (BaseSet` U)   &   |- S = (.s` U)   &   |- Z = (0v` U)   =>   |- ((U e. CHil /\ A e. X) -> (0SA) = Z)
Theorem	hlipf 8874	Mapping for Hilbert space inner product.
|- X = (BaseSet` U)   &   |- P = (.i` U)   =>   |- (U e. CHil -> P:(X X. X)-->CC)
Theorem	hlipcj 8875	Conjugate law for Hilbert space inner product.
|- X = (BaseSet` U)   &   |- P = (.i` U)   =>   |- ((U e. CHil /\ A e. X /\ B e. X) -> (APB) = (*` (BPA)))
Theorem	hlipdir 8876	Distributive law for Hilbert space inner product.
|- X = (BaseSet` U)   &   |- G = (+v` U)   &   |- P = (.i` U)   =>   |- ((U e. CHil /\ (A e. X /\ B e. X /\ C e. X)) -> ((AGB)PC) = ((APC) + (BPC)))
Theorem	hlipass 8877	Associative law for Hilbert space inner product.
|- X = (BaseSet` U)   &   |- S = (.s` U)   &   |- P = (.i` U)   =>   |- ((U e. CHil /\ (A e. CC /\ B e. X /\ C e. X)) -> ((ASB)PC) = (A x. (BPC)))
Theorem	hlipgt0 8878	The inner product of a Hilbert space vector by itself is positive.
|- X = (BaseSet` U)   &   |- Z = (0v` U)   &   |- P = (.i` U)   =>   |- ((U e. CHil /\ A e. X /\ A =/= Z) -> 0 < (APA))
Theorem	hlcompl 8879	Completeness of a Hilbert space.
|- X = (BaseSet` U)   &   |- D = (IndMet` U)   =>   |- ((U e. CHil /\ F e. (Cau` D)) -> E.x e. X F(~~>m` D)x)
Examples of complex Hilbert spaces
Theorem	cnhl 8880	The set of complex numbers is a complex Hilbert space. (Contributed by Steve Rodriguez, 28-Apr-2007.)
|- U = <.<. + , x. >., abs>.   =>   |- U e. CHil
Subspaces
Theorem	ssphl 8881	A Banach subspace of an inner product space is a Hilbert space.
|- H = (SubSp` U)   =>   |- ((U e. CPreHil /\ W e. H /\ W e. CBan) -> W e. CHil)
Hellinger-Toeplitz Theorem
Theorem	htthlem1 8882	Lemma for htthi 8894. Closure of values of an operator T on an auxiliary sequence of vectors f.
Theorem	htthlem2 8883	Lemma for htthi 8894. Elevate set variables to class variables in the self-adjoint hypothesis.
Theorem	htthlem3 8884	Lemma for htthi 8894. Construct an auxiliary sequence of functionals F` k from inner products of the given function T and auxiliary vector sequence f.
Theorem	htthlem4 8885	Lemma for htthi 8894. Value of a functional F` k.
Theorem	htthlem5 8886	Lemma for htthi 8894. Each F` k is a bounded linear functional (i.e. a bounded linear operator from the vector space to CC).
Theorem	htthlem6 8887	Lemma for htthi 8894. An upper bound of all F` k at a given vector A, when the norms of auxiliary vector sequence f are all 1 or less.
Theorem	htthlem7 8888	Lemma for htthi 8894. Convert upper bound in htthlem6 8887 to an existence condition.
Theorem	htthlem8 8889	Lemma for htthi 8894.
Theorem	htthlem9 8890	Lemma for htthi 8894.
Theorem	htthlem10 8891	Lemma for htthi 8894.
Theorem	htthlem11 8892	Lemma for htthi 8894. Use the Uniform Boundedness Theorem ubthi 8804 to show that the functional F` k is bounded.
Theorem	htthlem12 8893	Lemma for htthi 8894. Linear operator T is bounded.
Theorem	htthi 8894	Hellinger-Toeplitz Theorem: any self-adjoint linear operator defined on all of Hilbert space is bounded. Theorem 10.1-1 of [Kreyszig] p. 525. Discovered by E. Hellinger and O. Toeplitz in 1910, "it aroused both admiration and puzzlement since the theorem establishes a relation between properties of two different kinds, namely, the properties of being defined everywhere and being bounded."
|- X = (BaseSet` U)   &   |- P = (.i` U)   &   |- L = (U LnOp U)   &   |- B = (U BLnOp U)   &   |- U e. CHil   &   |- T e. L   &   |- ((x e. X /\ y e. X) -> ((T` x)Py) = (xP(T` y)))   =>   |- T e. B
Posets and lattices
Definition and basic properties
Syntax	cps 8895	Extend class notation with the class of all posets.
class Poset
Syntax	cspw 8896	Extend class notation with the supremum of an ordered set.
class supw
Syntax	cinf 8897	Extend class notation with the supremum of an ordered set.
class infw
Syntax	cjn 8898	Extend class notation with the join of two ordered sets.
class join
Syntax	cmee 8899	Extend class notation with the meet of two ordered sets.
class meet
Syntax	cla 8900	Extend class notation with the class of all lattices.
class Lat