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	isblo 8401	The predicate "is a bounded linear operator."
|- N = (UnormOpW)   &   |- L = (U LnOp W)   &   |- B = (U BLnOp W)   =>   |- ((U e. NrmCVec /\ W e. NrmCVec) -> (T e. B <-> (T e. L /\ (N` T) < +oo)))
Theorem	isblo2 8402	The predicate "is a bounded linear operator."
|- N = (UnormOpW)   &   |- L = (U LnOp W)   &   |- B = (U BLnOp W)   =>   |- ((U e. NrmCVec /\ W e. NrmCVec) -> (T e. B <-> (T e. L /\ (N` T) e. RR)))
Theorem	bloln 8403	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 8404	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 8405	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 8406	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 8407	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 8408	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 8409	The zero operator is linear.
|- Z = (U 0op W)   &   |- L = (U LnOp W)   =>   |- ((U e. NrmCVec /\ W e. NrmCVec) -> Z e. L)
Theorem	nmo0 8410	The operator norm of the zero operator.
|- N = (UnormOpW)   &   |- Z = (U 0op W)   =>   |- ((U e. NrmCVec /\ W e. NrmCVec) -> (N` Z) = 0)
Theorem	0blo 8411	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 8412	Lemma for nmlno0i 8413.
Theorem	nmlno0i 8413	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 8414	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 8415	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 8416	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 8417	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 8418	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 8419	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 8420	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 8421	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 8422	Lemma for blocni 8423 and lnocni 8424. If a linear operator is continuous at any point, it is bounded. Warning: The HTML proof page is 0.7MB in size.
Theorem	blocni 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)   &   |- 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 8424	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 8425	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 8426	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 8427	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 8428	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 8429	Extend class notation with the class of all complex inner product spaces (also called pre-Hilbert spaces).
class CPreHil
Definition	df-ph 8430	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>. |