mmtheorems88 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 8701-8800 - Page 88 of 123

Type	Label	Description
Statement
Theorem	nmblore 8701	The norm of a bounded operator is a real number.
|- X = (BaseSet` U)   &   |- Y = (BaseSet` W)   &   |- N = (UnormOpW)   &   |- B = (U BLnOp W)   =>   |- ((U e. NrmCVec /\ W e. NrmCVec /\ T e. B) -> (N` T) e. RR)
Theorem	0ofval 8702	The zero operator between two normed complex vector spaces.
|- X = (BaseSet` U)   &   |- Z = (0v` W)   &   |- O = (U 0op W)   =>   |- ((U e. NrmCVec /\ W e. NrmCVec) -> O = (X X. {Z}))
Theorem	0oval 8703	Value of the zero operator.
|- X = (BaseSet` U)   &   |- Z = (0v` W)   &   |- O = (U 0op W)   =>   |- ((U e. NrmCVec /\ W e. NrmCVec /\ A e. X) -> (O` A) = Z)
Theorem	0oo 8704	The zero operator is an operator.
|- X = (BaseSet` U)   &   |- Y = (BaseSet` W)   &   |- Z = (U 0op W)   =>   |- ((U e. NrmCVec /\ W e. NrmCVec) -> Z:X-->Y)
Theorem	0lno 8705	The zero operator is linear.
|- Z = (U 0op W)   &   |- L = (U LnOp W)   =>   |- ((U e. NrmCVec /\ W e. NrmCVec) -> Z e. L)
Theorem	nmo0 8706	The operator norm of the zero operator.
|- N = (UnormOpW)   &   |- Z = (U 0op W)   =>   |- ((U e. NrmCVec /\ W e. NrmCVec) -> (N` Z) = 0)
Theorem	0blo 8707	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 8708	Lemma for nmlno0i 8709.
Theorem	nmlno0i 8709	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 8710	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 8711	An upper bound for the operator norm of a linear operator, using only the properties of nonzero arguments.
|- X = (BaseSet` 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 8712	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	lnon0 8713	The domain of a nonzero linear operator contains a nonzero vector.
|- X = (BaseSet` U)   &   |- Z = (0v` U)   &   |- O = (U 0op W)   &   |- L = (U LnOp W)   =>   |- (((U e. NrmCVec /\ W e. NrmCVec /\ T e. L) /\ T =/= O) -> E.x e. X x =/= Z)
Theorem	nmblolbii 8714	A lower bound for the norm of a bounded linear operator.
|- X = (BaseSet` 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 8715	A lower bound for the norm of a bounded linear operator.
|- X = (BaseSet` 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 8716	The predicate "is a bounded linear operator." Definition 2.7-1 of [Kreyszig] p. 91.
|- X = (BaseSet` 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 8717	Properties that determine a bounded linear operator.
|- X = (BaseSet` 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 8718	Upper bound for the distance between the values of a bounded linear operator.
|- X = (BaseSet` U)   &   |- Y = (BaseSet` 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 8719	Lemma for blocni 8720 and lnocni 8721. If a linear operator is continuous at any point, it is bounded. Warning: The HTML proof page is 0.7MB in size.
Theorem	blocni 8720	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 8721	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 = (BaseSet` U)   =>   |- ((P e. X /\ T e. ((J CnP K)` P)) -> T e. (J Cn K))
Theorem	blocn 8722	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 8723	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 8724	The adjoint function.
|- X = (BaseSet` U)   &   |- Y = (BaseSet` 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 8725	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})
Theorem	ishmo 8726	The predicate "is a hermitian operator."
|- H = (HmOp` U)   &   |- A = (UadjU)   =>   |- (U e. NrmCVec -> (T e. H <-> (T e. dom A /\ (A` T) = T)))
Inner product (pre-Hilbert) spaces
Definition and basic properties
Syntax	cphl 8727	Extend class notation with the class of all complex inner product spaces (also called pre-Hilbert spaces).
class CPreHil
Definition	df-ph 8728	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 8729	Every complex inner product space is a normed complex vector space.
|- (U e. CPreHil -> U e. NrmCVec)
Theorem	phrel 8730	The class of all complex inner product spaces is a relation.
|- Rel CPreHil
Theorem	phnvi 8731	Every complex inner product space is a normed complex vector space.
|- U e. CPreHil   =>   |- U e. NrmCVec
Theorem	isphg 8732	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 G   =>   |- ((G e. A /\ S e. B /\ N e. C) -> (<.<.G, S>., N>. e. CPreHil <-> (<.<.G, S>., N>. e. NrmCVec /\ A.x e. X A.y e. X (((N` (xGy))^2) + ((N` (xG(-u1Sy)))^2)) = (2 x. (((N` x)^2) + ((N` y)^2))))))
Theorem	phop 8733	A complex inner product space in terms of ordered pair components.
|- G = (+v` U)   &   |- S = (.s` U)   &   |- N = (norm` U)   =>   |- (U e. CPreHil -> U = <.<.G, S>., N>.)
Examples of pre-Hilbert spaces
Theorem	cnph 8734	The set of complex numbers is an inner product (pre-Hilbert) space. (Contributed by Steve Rodriguez, 28-Apr-2007; revised by nm 24-Jul-2008.)
|- U = <.<. + , x. >., abs>.   =>   |- U e. CPreHil
Theorem	elimph 8735	Hypothesis elimination lemma for complex inner product spaces to assist weak deduction theorem.
|- X = (BaseSet` U)   &   |- Z = (0v` U)   &   |- U e. CPreHil   =>   |- if(A e. X, A, Z) e. X
Theorem	elimphu 8736	Hypothesis elimination lemma for complex inner product spaces to assist weak deduction theorem.
|- if(U e. CPreHil, U, <.<. + , x. >., abs>.) e. CPreHil
Properties of pre-Hilbert spaces
Theorem	isph 8737	The predicate "is an inner product space."
|- X = (BaseSet` U)   &   |- G = (+v` U)   &   |- M = (-v` U)   &   |- N = (norm` U)   =>   |- (U e. CPreHil <-> (U e. NrmCVec /\ A.x e. X A.y e. X (((N` (xGy))^2) + ((N` (xMy))^2)) = (2 x. (((N` x)^2) + ((N` y)^2)))))
Theorem	phpar2 8738	The parallelogram law for an inner product space.
|- X = (BaseSet` U)   &   |- G = (+v` U)   &   |- M = (-v` U)   &   |- N = (norm` U)   =>   |- ((U e. CPreHil /\ A e. X /\ B e. X) -> (((N` (AGB))^2) + ((N` (AMB))^2)) = (2 x. (((N` A)^2) + ((N` B)^2))))
Theorem	phpar 8739	The parallelogram law for an inner product space.
|- X = (BaseSet` U)   &   |- G = (+v` U)   &   |- S = (.s` U)   &   |- N = (norm` U)   =>   |- ((U e. CPreHil /\ A e. X /\ B e. X) -> (((N` (AGB))^2) + ((N` (AG(-u1SB)))^2)) = (2 x. (((N` A)^2) + ((N` B)^2))))
Theorem	ip0i 8740	A slight variant of Equation 6.46 of [Ponnusamy] p. 362, where J is either 1 or -1 to represent +-1.
|- X = (BaseSet` U)   &   |- G = (+v` U)   &   |- S = (.s` U)   &   |- P = (.i` U)   &   |- U e. CPreHil   &   |- A e. X   &   |- B e. X   &   |- C e. X   &   |- N = (norm` U)   &   |- J e. CC   =>   |- ((((N` ((AGB)G(JSC)))^2) - ((N` ((AGB)G(-uJSC)))^2)) + (((N` ((AG(-u1SB))G(JSC)))^2) - ((N` ((AG(-u1SB))G(-uJSC)))^2))) = (2 x. (((N` (AG(JSC)))^2) - ((N` (AG(-uJSC)))^2)))
Theorem	ip1ilem 8741	Lemma for ip1i 8742.
Theorem	ip1i 8742	Equation 6.47 of [Ponnusamy] p. 362.
|- X = (BaseSet` U)   &   |- G = (+v` U)   &   |- S = (.s` U)   &   |- P = (.i` U)   &   |- U e. CPreHil   &   |- A e. X   &   |- B e. X   &   |- C e. X   =>   |- (((AGB)PC) + ((AG(-u1SB))PC)) = (2 x. (APC))
Theorem	ip2i 8743	Equation 6.48 of [Ponnusamy] p. 362.
|- X = (BaseSet` U)   &   |- G = (+v` U)   &   |- S = (.s` U)   &   |- P = (.i` U)   &   |- U e. CPreHil   &   |- A e. X   &   |- B e. X   =>   |- ((2SA)PB) = (2 x. (APB))
Theorem	ipdirilem 8744	Lemma for ipdiri 8745.
Theorem	ipdiri 8745	Distributive law for inner product. Equation I3 of [Ponnusamy] p. 362.
|- X = (BaseSet` U)   &   |- G = (+v` U)   &   |- S = (.s` U)   &   |- P = (.i` U)   &   |- U e. CPreHil   =>   |- ((A e. X /\ B e. X /\ C e. X) -> ((AGB)PC) = ((APC) + (BPC)))
Theorem	ipasslem1 8746	Lemma for ipassi 8757. Show the inner product associative law for nonnegative integers.
Theorem	ipasslem2 8747	Lemma for ipassi 8757. Show the inner product associative law for nonpositive integers.
Theorem	ipasslem3 8748	Lemma for ipassi 8757. Show the inner product associative law for all integers.
Theorem	ipasslem4 8749	Lemma for ipassi 8757. Show the inner product associative law for positive integer reciprocals.
Theorem	ipasslem5 8750	Lemma for ipassi 8757. Show the inner product associative law for rational numbers.
Theorem	ipasslem6 8751	Lemma for ipassi 8757. Show that ((wSA)PB) - (w x. (APB)) is continuous in w.
Theorem	ipasslem7 8752	Lemma for ipassi 8757. Show that ((wSA)PB) - (w x. (APB)) is continuous on RR.
Theorem	ipasslem8 8753	Lemma for ipassi 8757. By ipasslem5 8750, F is 0 for all QQ; since it is continuous and QQ is dense in RR by qdensere2 8127, we conclude F is 0 for all RR.
Theorem	ipasslem9 8754	Lemma for ipassi 8757. Conclude from ipasslem8 8753 the inner product associative law for real numbers.
Theorem	ipasslem10 8755	Lemma for ipassi 8757. Show the inner product associative law for the imaginary number i.
Theorem	ipasslem11 8756	Lemma for ipassi 8757. Show the inner product associative law for all complex numbers.
Theorem	ipassi 8757	Associative law for inner product. Equation I2 of [Ponnusamy] p. 363.
|- X = (BaseSet` U)   &   |- G = (+v` U)   &   |- S = (.s` U)   &   |- P = (.i` U)   &   |- U e. CPreHil   =>   |- ((A e. CC /\ B e. X /\ C e. X) -> ((ASB)PC) = (A x. (BPC)))
Theorem	ipdir 8758	Distributive law for inner product. Equation I3 of [Ponnusamy] p. 362.
|- X = (BaseSet` U)   &   |- G = (+v` U)   &   |- P = (.i` U)   =>   |- ((U e. CPreHil /\ (A e. X /\ B e. X /\ C e. X)) -> ((AGB)PC) = ((APC) + (BPC)))
Theorem	ipdi 8759	Distributive law for inner product.
|- X = (BaseSet` U)   &   |- G = (+v` U)   &   |- P = (.i` U)   =>   |- ((U e. CPreHil /\ (A e. X /\ B e. X /\ C e. X)) -> (AP(BGC)) = ((APB) + (APC)))
Theorem	ip2dii 8760	Inner product of two sums.
|- X = (BaseSet` U)   &   |- G = (+v` U)   &   |- P = (.i` U)   &   |- U e. CPreHil   &   |- A e. X   &   |- B e. X   &   |- C e. X   &   |- D e. X   =>   |- ((AGB)P(CGD)) = (((APC) + (BPD)) + ((APD) + (BPC)))
Theorem	ipass 8761	Associative law for inner product. Equation I2 of [Ponnusamy] p. 363.
|- X = (BaseSet` U)   &   |- S = (.s` U)   &   |- P = (.i` U)   =>   |- ((U e. CPreHil /\ (A e. CC /\ B e. X /\ C e. X)) -> ((ASB)PC) = (A x. (BPC)))
Theorem	ipassr 8762	"Associative" law for second argument of inner product (compare ipass 8761).
|- X = (BaseSet` U)   &   |- S = (.s` U)   &   |- P = (.i` U)   =>   |- ((U e. CPreHil /\ (A e. X /\ B e. CC /\ C e. X)) -> (AP(BSC)) = ((*` B) x. (APC)))
Theorem	ipassr2 8763	"Associative" law for inner product. Conjugate version of ipassr 8762.
|- X = (BaseSet` U)   &   |- S = (.s` U)   &   |- P = (.i` U)   =>   |- ((U e. CPreHil /\ (A e. X /\ B e. CC /\ C e. X)) -> (AP((*` B)SC)) = (B x. (APC)))
Theorem	ipsubdir 8764	Distributive law for inner product subtraction.
|- X = (BaseSet` U)   &   |- M = (-v` U)   &   |- P = (.i` U)   =>   |- ((U e. CPreHil /\ (A e. X /\ B e. X /\ C e. X)) -> ((AMB)PC) = ((APC) - (BPC)))
Theorem	ipsubdi 8765	Distributive law for inner product subtraction.
|- X = (BaseSet` U)   &   |- M = (-v` U)   &   |- P = (.i` U)   =>   |- ((U e. CPreHil /\ (A e. X /\ B e. X /\ C e. X)) -> (AP(BMC)) = ((APB) - (APC)))
Theorem	pythi 8766	The Pythagorean theorem for an arbitrary complex inner product (pre-Hilbert) space U. The square of the norm of the sum of two orthogonal vectors (i.e. whose inner product is 0) is the sum of the squares of their norms. Problem 2 in [Kreyszig] p. 135.
|- X = (BaseSet` U)   &   |- G = (+v` U)   &   |- N = (norm` U)   &   |- P = (.i` U)   &   |- U e. CPreHil   &   |- A e. X   &   |- B e. X   =>   |- ((APB) = 0 -> ((N` (AGB))^2) = (((N` A)^2) + ((N` B)^2)))
Theorem	siilem1 8767	Lemma for sii 8770.
Theorem	siilem2 8768	Lemma for sii 8770.
Theorem	siii 8769	Inference from sii 8770.
|- X = (BaseSet` U)   &   |- N = (norm` U)   &   |- P = (.i` U)   &   |- U e. CPreHil   &   |- A e. X   &   |- B e. X   =>   |- (abs` (APB)) <_ ((N` A) x. (N` B))
Theorem	sii 8770	Schwartz inequality. Part of Lemma 3-2.1(a) of [Kreyszig] p. 137.
|- X = (BaseSet` U)   &   |- N = (norm` U)   &   |- P = (.i` U)   &   |- U e. CPreHil   =>   |- ((A e. X /\ B e. X) -> (abs` (APB)) <_ ((N` A) x. (N` B)))
Theorem	sspph 8771	A subspace of an inner product space is an inner product space.
|- H = (SubSp` U)   =>   |- ((U e. CPreHil /\ W e. H) -> W e. CPreHil)
Theorem	ipblnfi 8772	A function F generated by varying the first argument of an inner product (with its second argument a fixed vector A) is a bounded linear functional, i.e. a bounded linear operator from the vector space to CC.
|- X = (BaseSet` U)   &   |- P = (.i` U)   &   |- U e. CPreHil   &   |- C = <.<. + , x. >., abs>.   &   |- B = (U BLnOp C)   &   |- F = {<.x, y>. | (x e. X /\ y = (xPA))}   =>   |- (A e. X -> F e. B)
Theorem	ip2eqi 8773	Two vectors are equal iff their inner products with all other vectors are equal.
|- X = (BaseSet` U)   &   |- P = (.i` U)   &   |- U e. CPreHil   =>   |- ((A e. X /\ B e. X) -> (A.x e. X (xPA) = (xPB) <-> A = B))
Theorem	phoeqi 8774	A condition implying that two operators are equal.
|- X = (BaseSet` U)   &   |- P = (.i` U)   &   |- U e. CPreHil   =>   |- ((S:Y-->X /\ T:Y-->X) -> (A.x e. X A.y e. Y (xP(S` y)) = (xP(T` y)) <-> S = T))
Theorem	ajmoi 8775	Every operator has at most one adjoint.
|- X = (BaseSet` U)   &   |- P = (.i` U)   &   |- U e. CPreHil   =>   |- E*s(s:Y-->X /\ A.x e. X A.y e. Y ((T` x)Qy) = (xP(s` y)))
Theorem	ajfuni 8776	The adjoint function is a function.
|- A = (UadjW)   &   |- U e. CPreHil   &   |- W e. NrmCVec   =>   |- Fun A
Theorem	ajfun 8777	The adjoint function is a function. This is not immediately apparent from df-aj 8665 but results from the uniqueness shown by ajmoi 8775.
|- A = (UadjW)   =>   |- ((U e. CPreHil /\ W e. NrmCVec) -> Fun A)
Theorem	ajval 8778	Value of the adjoint function.
|- X = (BaseSet` U)   &   |- Y = (BaseSet` W)   &   |- P = (.i` U)   &   |- Q = (.i` W)   &   |- A = (UadjW)   =>   |- ((U e. CPreHil /\ W e. NrmCVec /\ T:X-->Y) -> (A` T) = U.{s | (s:Y-->X /\ A.x e. X A.y e. Y ((T` x)Qy) = (xP(s` y)))})
Complex Banach spaces
Definition and basic properties
Syntax	cbn 8779	Extend class notation with the class of all complex Banach spaces.
class CBan
Definition	df-bn 8780	Define the class of all complex Banach spaces.
|- CBan = {u e. NrmCVec | (IndMet` u) e. CMet}
Theorem	isbn 8781	A complex Banach space is a normed complex vector space with a complete induced metric.
|- D = (IndMet` U)   =>   |- (U e. CBan <-> (U e. NrmCVec /\ D e. CMet))
Theorem	bncms 8782	The induced metric on complex Banach space is complete.
|- D = (IndMet` U)   =>   |- (U e. CBan -> D e. CMet)
Theorem	bnnv 8783	Every complex Banach space is a normed complex vector space.
|- (U e. CBan -> U e. NrmCVec)
Theorem	bnrel 8784	The class of all complex Banach spaces is a relation.
|- Rel CBan
Theorem	bnsscmcl 8785	A subspace of a Banach space is a Banach space iff it is closed in the norm-induced metric of the parent space.
|- X = (BaseSet` U)   &   |- D = (IndMet` U)   &   |- J = (Open` D)   &   |- H = (SubSp` U)   &   |- Y = (BaseSet` W)   =>   |- ((U e. CBan /\ W e. H) -> (W e. CBan <-> Y e. (Clsd` J)))
Examples of complex Banach spaces
Theorem	cnbn 8786	The set of complex numbers is a complex Banach space. (Contributed by Steve Rodriguez, 4-Jan-2007.)
|- U = <.<. + , x. >., abs>.   =>   |- U e. CBan
Uniform Boundedness Theorem
Theorem	ubthlem1 8787	Lemma for ubthi 8804. Membership in A` k, the set of all vectors (T` n)` z whose norm is less than k.
Theorem	ubthlem2 8788	Lemma for ubthi 8804. A` k is a set of vectors.
Theorem	ubthlem3 8789	Lemma for ubthi 8804. The limit of any sequence in A` k has operator value norm less than k, for any bounded linear operator T` n, using metcn4i 8183 (with continuity shown by blocn2 8723) and the properties of A` k.
Theorem	ubthlem4 8790	Lemma for ubthi 8804. The set A` k is therefore closed by metcld 8178. (The hypothesis U e. CBan could be weakened to U e. NrmCVec.)
Theorem	ubthlem5 8791	Lemma for ubthi 8804. The union of all A` k equals the base set, provided the values of the operators T` n are bounded for each vector x.
Theorem	ubthlem6 8792	Lemma for ubthi 8804. Using bcth 8243 (via bcthlem33 8242), at least one set A` k contains a ball.
Theorem	ubthlem7 8793	Lemma for ubthi 8804. Auxiliary class Q is a vector.
Theorem	ubthlem8 8794	Lemma for ubthi 8804. Compute x in terms of auxiliary vector Q.
Theorem	ubthlem9 8795	Lemma for ubthi 8804. Evaluate the operator value at x in terms of the operator value at Q - p.
Theorem	ubthlem10 8796	Lemma for ubthi 8804. Upper limit for the norm of an operator value at auxiliary vector Q.
Theorem	ubthlem11 8797	Lemma for ubthi 8804. Upper limit for the norm of an operator value at Q - p.
Theorem	ubthlem12 8798	Lemma for ubthi 8804. Upper limit for the norm of an operator value at x.
Theorem	ubthlem12OLD 8799	Lemma for ubthi 8804. Upper limit for the norm of an operator value at x.
Theorem	ubthlem13 8800	Lemma for ubthi 8804. Upper bound for the operator norm of any operator T` n.