mmtheorems85 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 8401-8500 - Page 85 of 107

Type	Label	Description
Statement
Theorem	bloln 8401	A bounded operator is a linear operator.
|- L = (U LnOp W)   &   |- B = (U BLnOp W)   =>   |- ((U e. NrmCVec /\ W e. NrmCVec /\ T e. B) -> T e. L)
Theorem	blof 8402	A bounded operator is an operator.
|- X = (Base` U)   &   |- Y = (Base` W)   &   |- B = (U BLnOp W)   =>   |- ((U e. NrmCVec /\ W e. NrmCVec /\ T e. B) -> T:X-->Y)
Theorem	nmblore 8403	The norm of a bounded operator is a real number.
|- X = (Base` U)   &   |- Y = (Base` W)   &   |- N = (UnormOpW)   &   |- B = (U BLnOp W)   =>   |- ((U e. NrmCVec /\ W e. NrmCVec /\ T e. B) -> (N` T) e. RR)
Theorem	0ofval 8404	The zero operator between two normed complex vector spaces.
|- X = (Base` U)   &   |- Z = (0v` W)   &   |- O = (U 0op W)   =>   |- ((U e. NrmCVec /\ W e. NrmCVec) -> O = (X X. {Z}))
Theorem	0oval 8405	Value of the zero operator.
|- X = (Base` U)   &   |- Z = (0v` W)   &   |- O = (U 0op W)   =>   |- ((U e. NrmCVec /\ W e. NrmCVec /\ A e. X) -> (O` A) = Z)
Theorem	0oo 8406	The zero operator is an operator.
|- X = (Base` U)   &   |- Y = (Base` W)   &   |- Z = (U 0op W)   =>   |- ((U e. NrmCVec /\ W e. NrmCVec) -> Z:X-->Y)
Theorem	0lno 8407	The zero operator is linear.
|- Z = (U 0op W)   &   |- L = (U LnOp W)   =>   |- ((U e. NrmCVec /\ W e. NrmCVec) -> Z e. L)
Theorem	nmo0 8408	The operator norm of the zero operator.
|- N = (UnormOpW)   &   |- Z = (U 0op W)   =>   |- ((U e. NrmCVec /\ W e. NrmCVec) -> (N` Z) = 0)
Theorem	0blo 8409	The zero operator is a bounded linear operator.
|- Z = (U 0op W)   &   |- B = (U BLnOp W)   =>   |- ((U e. NrmCVec /\ W e. NrmCVec) -> Z e. B)
Theorem	nmlno0lem 8410	Lemma for nmlno0i 8411.
Theorem	nmlno0i 8411	The norm of a linear operator is zero iff the operator is zero.
|- N = (UnormOpW)   &   |- Z = (U 0op W)   &   |- L = (U LnOp W)   &   |- U e. NrmCVec   &   |- W e. NrmCVec   =>   |- (T e. L -> ((N` T) = 0 <-> T = Z))
Theorem	nmlno0 8412	The norm of a linear operator is zero iff the operator is zero.
|- N = (UnormOpW)   &   |- Z = (U 0op W)   &   |- L = (U LnOp W)   =>   |- ((U e. NrmCVec /\ W e. NrmCVec /\ T e. L) -> ((N` T) = 0 <-> T = Z))
Theorem	nmlnoubi 8413	An upper bound for the operator norm of a linear operator, using only the properties of nonzero arguments.
|- X = (Base` U)   &   |- Z = (0v` U)   &   |- K = (norm` U)   &   |- M = (norm` W)   &   |- N = (UnormOpW)   &   |- L = (U LnOp W)   &   |- U e. NrmCVec   &   |- W e. NrmCVec   =>   |- ((T e. L /\ (A e. RR /\ 0 <_ A) /\ A.x e. X (x =/= Z -> (M` (T` x)) <_ (A x. (K` x)))) -> (N` T) <_ A)
Theorem	nmlnogt0 8414	The norm of a nonzero linear operator is positive.
|- N = (UnormOpW)   &   |- Z = (U 0op W)   &   |- L = (U LnOp W)   =>   |- ((U e. NrmCVec /\ W e. NrmCVec /\ T e. L) -> (T =/= Z <-> 0 < (N` T)))
Theorem	nmblolbii 8415	A lower bound for the norm of a bounded linear operator.
|- X = (Base` U)   &   |- L = (norm` U)   &   |- M = (norm` W)   &   |- N = (UnormOpW)   &   |- B = (U BLnOp W)   &   |- U e. NrmCVec   &   |- W e. NrmCVec   &   |- T e. B   =>   |- (A e. X -> (M` (T` A)) <_ ((N` T) x. (L` A)))
Theorem	nmblolbi 8416	A lower bound for the norm of a bounded linear operator.
|- X = (Base` U)   &   |- L = (norm` U)   &   |- M = (norm` W)   &   |- N = (UnormOpW)   &   |- B = (U BLnOp W)   &   |- U e. NrmCVec   &   |- W e. NrmCVec   =>   |- ((T e. B /\ A e. X) -> (M` (T` A)) <_ ((N` T) x. (L` A)))
Theorem	isblo3i 8417	The predicate "is a bounded linear operator." Definition 2.7-1 of [Kreyszig] p. 91.
|- X = (Base` U)   &   |- M = (norm` U)   &   |- N = (norm` W)   &   |- L = (U LnOp W)   &   |- B = (U BLnOp W)   &   |- U e. NrmCVec   &   |- W e. NrmCVec   =>   |- (T e. B <-> (T e. L /\ E.x e. RR A.y e. X (N` (T` y)) <_ (x x. (M` y))))
Theorem	blo3i 8418	Properties that determine a bounded linear operator.
|- X = (Base` U)   &   |- M = (norm` U)   &   |- N = (norm` W)   &   |- L = (U LnOp W)   &   |- B = (U BLnOp W)   &   |- U e. NrmCVec   &   |- W e. NrmCVec   =>   |- ((T e. L /\ A e. RR /\ A.y e. X (N` (T` y)) <_ (A x. (M` y))) -> T e. B)
Theorem	blometi 8419	Upper bound for the distance between the values of a bounded linear operator.
|- X = (Base` U)   &   |- Y = (Base` W)   &   |- C = (IndMet` U)   &   |- D = (IndMet` W)   &   |- N = (UnormOpW)   &   |- B = (U BLnOp W)   &   |- U e. NrmCVec   &   |- W e. NrmCVec   =>   |- ((T e. B /\ P e. X /\ Q e. X) -> ((T` P)D(T` Q)) <_ ((N` T) x. (PCQ)))
Theorem	blocnilem 8420	Lemma for blocni 8421 and lnocni 8422. If a linear operator is continuous at any point, it is bounded. Warning: The HTML proof page is 0.7MB in size.
Theorem	blocni 8421	A linear operator is continuous iff it is bounded. Theorem 2.7-9(a) of [Kreyszig] p. 97.
|- C = (IndMet` U)   &   |- D = (IndMet` W)   &   |- J = (Open` C)   &   |- K = (Open` D)   &   |- L = (U LnOp W)   &   |- B = (U BLnOp W)   &   |- U e. NrmCVec   &   |- W e. NrmCVec   &   |- T e. L   =>   |- (T e. (J Cn K) <-> T e. B)
Theorem	lnocni 8422	If a linear operator is continuous at any point, it is continuous everywhere. Theorem 2.7-9(b) of [Kreyszig] p. 97.
|- C = (IndMet` U)   &   |- D = (IndMet` W)   &   |- J = (Open` C)   &   |- K = (Open` D)   &   |- L = (U LnOp W)   &   |- B = (U BLnOp W)   &   |- U e. NrmCVec   &   |- W e. NrmCVec   &   |- T e. L   &   |- X = (Base` U)   =>   |- ((P e. X /\ T e. ((J CnP K)` P)) -> T e. (J Cn K))
Theorem	blocn 8423	A linear operator is continuous iff it is bounded. Theorem 2.7-9(a) of [Kreyszig] p. 97.
|- C = (IndMet` U)   &   |- D = (IndMet` W)   &   |- J = (Open` C)   &   |- K = (Open` D)   &   |- B = (U BLnOp W)   &   |- U e. NrmCVec   &   |- W e. NrmCVec   &   |- L = (U LnOp W)   =>   |- (T e. L -> (T e. (J Cn K) <-> T e. B))
Theorem	blocn2 8424	A bounded linear operator is continuous.
|- C = (IndMet` U)   &   |- D = (IndMet` W)   &   |- J = (Open` C)   &   |- K = (Open` D)   &   |- B = (U BLnOp W)   &   |- U e. NrmCVec   &   |- W e. NrmCVec   =>   |- (T e. B -> T e. (J Cn K))
Theorem	ajfval 8425	The adjoint function.
|- X = (Base` U)   &   |- Y = (Base` W)   &   |- P = (.i` U)   &   |- Q = (.i` W)   &   |- A = (UadjW)   =>   |- ((U e. NrmCVec /\ W e. NrmCVec) -> A = {<.t, s>. | (t:X-->Y /\ s:Y-->X /\ A.x e. X A.y e. Y ((t` x)Qy) = (xP(s` y)))})
Theorem	hmoval 8426	The set of Hermitian (self-adjoint) operators on a normed complex vector space.
|- H = (HmOp` U)   &   |- A = (UadjU)   =>   |- (U e. NrmCVec -> H = {t e. dom A | (A` t) = t})
Inner product (pre-Hilbert) spaces
Definition and basic properties
Syntax	cphl 8427	Extend class notation with the class of all complex inner product spaces (also called pre-Hilbert spaces).
class CPreHil
Definition	df-ph 8428	Define the class of all complex inner product spaces. An inner product space is a normed vector space whose norm satisfies the parallelogram law (a property that induces an inner product). Based on Exercise 4(b) of [ReedSimon] p. 63. The vector operation is g, the scalar product is s, and the norm is n. An inner product space is also called a pre-Hilbert space.
|- CPreHil = (NrmCVec i^i {<.<.g, s>., n>. | A.x e. ran gA.y e. ran g(((n` (xgy))^2) + ((n` (xg(-u1sy)))^2)) = (2 x. (((n` x)^2) + ((n` y)^2)))})
Theorem	phnv 8429	Every complex inner product space is a normed complex vector space.
|- (U e. CPreHil -> U e. NrmCVec)
Theorem	phrel 8430	The class of all complex inner product spaces is a relation.
|- Rel CPreHil
Theorem	phnvi 8431	Every complex inner product space is a normed complex vector space.
|- U e. CPreHil   =>   |- U e. NrmCVec
Theorem	isphg 8432	The predicate "is a complex inner product space." An inner product space is a normed vector space whose norm satisfies the parallelogram law. The vector (group) addition operation is G, the scalar product is S, and the norm is N. An inner product space is also called a pre-Hilbert space.
|- X = ran