mmtheorems87 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 8601-8700 - Page 87 of 123

Type	Label	Description
Statement
Theorem	sm1cnilem 8601	Lemma for sm1cni 8602.
Theorem	sm1cni 8602	Scalar multiplication is continuous in its first operand.
|- X = (BaseSet` U)   &   |- S = (.s` U)   &   |- C = (abs o. - )   &   |- D = (IndMet` U)   &   |- J = (Open` C)   &   |- K = (Open` D)   &   |- F = {<.w, v>. | (w e. CC /\ v = (wSA))}   &   |- U e. NrmCVec   &   |- A e. X   =>   |- F e. (J Cn K)
Inner product
Syntax	cip 8603	Extend class notation with the class inner product functions.
class .i
Definition	df-ip 8604	Define a function that maps a complex normed vector space to its inner product operation in case its norm satisfies the parallelogram identity (otherwise the operation is still defined, but not meaningful). Based on Exercise 4(a) of [ReedSimon] p. 63 and Theorem 6.44 of [Ponnusamy] p. 361. Vector addition is (1st` w), the scalar product is (2nd` w), and the norm is n.
|- .i = {<.<.w, n>., p>. | (<.w, n>. e. NrmCVec /\ p = {<.<.x, y>., z>. | ((x e. dom n /\ y e. dom n) /\ z = (sum_k e. (1...4)((i^k) x. ((n` (x(1st` w)((i^k)(2nd` w)y)))^2)) / 4))})}
Theorem	ipval2lem1 8605	Lemma for ipval3 8613.
Theorem	ipfval 8606	The inner product function on a normed complex vector space. The definition is meaningful for vector spaces that are also inner product spaces, i.e. satisfy the parallelogram law.
|- X = (BaseSet` U)   &   |- G = (+v` U)   &   |- S = (.s` U)   &   |- N = (norm` U)   &   |- P = (.i` U)   =>   |- (U e. NrmCVec -> P = {<.<.x, y>., z>. | ((x e. X /\ y e. X) /\ z = (sum_k e. (1...4)((i^k) x. ((N` (xG((i^k)Sy)))^2)) / 4))})
Theorem	ipval 8607	Value of the inner product. The definition is meaningful for normed complex vector spaces that are also inner product spaces, i.e. satisfy the parallelogram law, although for convenience we define it for any normed complex vector space. The vector (group) addition operation is G, the scalar product is S, the norm is N, and the set of vectors is X. Equation 6.45 of [Ponnusamy] p. 361.
|- X = (BaseSet` U)   &   |- G = (+v` U)   &   |- S = (.s` U)   &   |- N = (norm` U)   &   |- P = (.i` U)   =>   |- ((U e. NrmCVec /\ A e. X /\ B e. X) -> (APB) = (sum_k e. (1...4)((i^k) x. ((N` (AG((i^k)SB)))^2)) / 4))
Theorem	ipval2lem2 8608	Lemma for ipval3 8613.
Theorem	ipval2lem3 8609	Lemma for ipval3 8613.
Theorem	ipval2lem4 8610	Lemma for ipval3 8613.
Theorem	ipval2 8611	Expansion of the inner product value ipval 8607. Warning: The HTML proof page is 0.5MB in size.
|- X = (BaseSet` U)   &   |- G = (+v` U)   &   |- S = (.s` U)   &   |- N = (norm` U)   &   |- P = (.i` U)   =>   |- ((U e. NrmCVec /\ A e. X /\ B e. X) -> (APB) = (((((N` (AGB))^2) - ((N` (AG(-u1SB)))^2)) + (i x. (((N` (AG(iSB)))^2) - ((N` (AG(-uiSB)))^2)))) / 4))
Theorem	4ipval2 8612	Four times the inner product value ipval3 8613, useful for simplifying certain proofs.
|- X = (BaseSet` U)   &   |- G = (+v` U)   &   |- S = (.s` U)   &   |- N = (norm` U)   &   |- P = (.i` U)   =>   |- ((U e. NrmCVec /\ A e. X /\ B e. X) -> (4 x. (APB)) = ((((N` (AGB))^2) - ((N` (AG(-u1SB)))^2)) + (i x. (((N` (AG(iSB)))^2) - ((N` (AG(-uiSB)))^2)))))
Theorem	ipval3 8613	Expansion of the inner product value ipval 8607.
|- X = (BaseSet` U)   &   |- G = (+v` U)   &   |- S = (.s` U)   &   |- N = (norm` U)   &   |- P = (.i` U)   &   |- M = (-v` U)   =>   |- ((U e. NrmCVec /\ A e. X /\ B e. X) -> (APB) = (((((N` (AGB))^2) - ((N` (AMB))^2)) + (i x. (((N` (AG(iSB)))^2) - ((N` (AM(iSB)))^2)))) / 4))
Theorem	ipval2lem5 8614	Lemma for ipval3 8613.
Theorem	ipval2lem6 8615	Lemma for ipval3 8613.
Theorem	4ipval3 8616	Four times the inner product value ipval3 8613, useful for simplifying certain proofs.
|- X = (BaseSet` U)   &   |- G = (+v` U)   &   |- S = (.s` U)   &   |- N = (norm` U)   &   |- P = (.i` U)   &   |- M = (-v` U)   =>   |- ((U e. NrmCVec /\ A e. X /\ B e. X) -> (4 x. (APB)) = ((((N` (AGB))^2) - ((N` (AMB))^2)) + (i x. (((N` (AG(iSB)))^2) - ((N` (AM(iSB)))^2)))))
Theorem	ipid 8617	The inner product of a vector with itself is the square of the vector's norm. Equation I4 of [Ponnusamy] p. 362.
|- X = (BaseSet` U)   &   |- N = (norm` U)   &   |- P = (.i` U)   =>   |- ((U e. NrmCVec /\ A e. X) -> (APA) = ((N` A)^2))
Theorem	ipnm 8618	Norm expressed in terms of inner product.
|- X = (BaseSet` U)   &   |- N = (norm` U)   &   |- P = (.i` U)   =>   |- ((U e. NrmCVec /\ A e. X) -> (N` A) = (sqr` (APA)))
Theorem	ipcl 8619	An inner product is a complex number.
|- X = (BaseSet` U)   &   |- P = (.i` U)   =>   |- ((U e. NrmCVec /\ A e. X /\ B e. X) -> (APB) e. CC)
Theorem	ipf 8620	Mapping for the inner product operation.
|- X = (BaseSet` U)   &   |- P = (.i` U)   =>   |- (U e. NrmCVec -> P:(X X. X)-->CC)
Theorem	ipcj 8621	The complex conjugate of an inner product reverses its arguments. Equation I1 of [Ponnusamy] p. 362.
|- X = (BaseSet` U)   &   |- P = (.i` U)   =>   |- ((U e. NrmCVec /\ A e. X /\ B e. X) -> (*` (APB)) = (BPA))
Theorem	ipipcj 8622	An inner product times its conjugate.
|- X = (BaseSet` U)   &   |- P = (.i` U)   =>   |- ((U e. NrmCVec /\ A e. X /\ B e. X) -> ((APB) x. (BPA)) = ((abs` (APB))^2))
Theorem	iporthcom 8623	Orthogonality (meaning inner product is 0) is commutative.
|- X = (BaseSet` U)   &   |- P = (.i` U)   =>   |- ((U e. NrmCVec /\ A e. X /\ B e. X) -> ((APB) = 0 <-> (BPA) = 0))
Theorem	ip0r 8624	Inner product with a zero second argument.
|- X = (BaseSet` U)   &   |- Z = (0v` U)   &   |- P = (.i` U)   =>   |- ((U e. NrmCVec /\ A e. X) -> (APZ) = 0)
Theorem	ip0l 8625	Inner product with a zero first argument. Part of proof of Theorem 6.44 of [Ponnusamy] p. 361.
|- X = (BaseSet` U)   &   |- Z = (0v` U)   &   |- P = (.i` U)   =>   |- ((U e. NrmCVec /\ A e. X) -> (ZPA) = 0)
Theorem	ipz 8626	The inner product of a vector with itself is zero iff the vector is zero. Part of Definition 3.1-1 of [Kreyszig] p. 129.
|- X = (BaseSet` U)   &   |- Z = (0v` U)   &   |- P = (.i` U)   =>   |- ((U e. NrmCVec /\ A e. X) -> ((APA) = 0 <-> A = Z))
Theorem	ip1cnilem1 8627	Lemma for ip1cni 8633.
Theorem	ip1cnilem2 8628	Lemma for ip1cni 8633.
Theorem	ip1cnilem3 8629	Lemma for ip1cni 8633.
Theorem	ip1cnilem4 8630	Lemma for ip1cni 8633.
Theorem	ip1cnilem5 8631	Lemma for ip1cni 8633.
Theorem	ip1cnilem6 8632	Lemma for ip1cni 8633.
Theorem	ip1cni 8633	Inner product is continuous in its first operand.
|- X = (BaseSet` U)   &   |- G = (+v` U)   &   |- P = (.i` U)   &   |- C = (IndMet` U)   &   |- D = (abs o. - )   &   |- J = (Open` C)   &   |- K = (Open` D)   &   |- F = {<.w, v>. | (w e. X /\ v = (wPA))}   &   |- U e. NrmCVec   &   |- A e. X   =>   |- F e. (J Cn K)
Subspaces
Syntax	css 8634	Extend class notation with the class of all subspaces of complex normed vector spaces.
class SubSp
Definition	df-ssp 8635	Define the class of all subspaces of complex normed vector spaces.
|- SubSp = {<.u, s>. | (u e. NrmCVec /\ s = {w e. NrmCVec | ((+v` w) (_ (+v` u) /\ (.s` w) (_ (.s` u) /\ (norm` w) (_ (norm` u))})}
Theorem	sspval 8636	The set of all subspaces of a normed complex vector space.
|- G = (+v` U)   &   |- S = (.s` U)   &   |- N = (norm` U)   &   |- H = (SubSp` U)   =>   |- (U e. NrmCVec -> H = {w e. NrmCVec | ((+v` w) (_ G /\ (.s` w) (_ S /\ (norm` w) (_ N)})
Theorem	isssp 8637	The predicate "is a subspace."
|- G = (+v` U)   &   |- F = (+v` W)   &   |- S = (.s` U)   &   |- R = (.s` W)   &   |- N = (norm` U)   &   |- M = (norm` W)   &   |- H = (SubSp` U)   =>   |- (U e. NrmCVec -> (W e. H <-> (W e. NrmCVec /\ (F (_ G /\ R (_ S /\ M (_ N))))
Theorem	sspid 8638	A normed complex vector space is a subspace of itself.
|- H = (SubSp` U)   =>   |- (U e. NrmCVec -> U e. H)
Theorem	sspnv 8639	A subspace is a normed complex vector space.
|- H = (SubSp` U)   =>   |- ((U e. NrmCVec /\ W e. H) -> W e. NrmCVec)
Theorem	sspba 8640	The base set of a subspace is included in the parent base set.
|- X = (BaseSet` U)   &   |- Y = (BaseSet` W)   &   |- H = (SubSp` U)   =>   |- ((U e. NrmCVec /\ W e. H) -> Y (_ X)
Theorem	sspg 8641	Vector addition on a subspace is a restriction of vector addition on the parent space.
|- Y = (BaseSet` W)   &   |- G = (+v` U)   &   |- F = (+v` W)   &   |- H = (SubSp` U)   =>   |- ((U e. NrmCVec /\ W e. H) -> F = (G |` (Y X. Y)))
Theorem	sspgval 8642	Vector addition on a subspace in terms of vector addition on the parent space.
|- Y = (BaseSet` W)   &   |- G = (+v` U)   &   |- F = (+v` W)   &   |- H = (SubSp` U)   =>   |- (((U e. NrmCVec /\ W e. H) /\ (A e. Y /\ B e. Y)) -> (AFB) = (AGB))
Theorem	ssps 8643	Scalar multiplication on a subspace is a restriction of scalar multiplication on the parent space.
|- Y = (BaseSet` W)   &   |- S = (.s` U)   &   |- R = (.s` W)   &   |- H = (SubSp` U)   =>   |- ((U e. NrmCVec /\ W e. H) -> R = (S |` (CC X. Y)))
Theorem	sspsval 8644	Scalar multiplication on a subspace in terms of scalar multiplication on the parent space.
|- Y = (BaseSet` W)   &   |- S = (.s` U)   &   |- R = (.s` W)   &   |- H = (SubSp` U)   =>   |- (((U e. NrmCVec /\ W e. H) /\ (A e. CC /\ B e. Y)) -> (ARB) = (ASB))
Theorem	sspmlem 8645	Lemma for sspm 8647 and others.
Theorem	sspmval 8646	Vector addition on a subspace in terms of vector addition on the parent space.
|- Y = (BaseSet` W)   &   |- M = (-v` U)   &   |- L = (-v` W)   &   |- H = (SubSp` U)   =>   |- (((U e. NrmCVec /\ W e. H) /\ (A e. Y /\ B e. Y)) -> (ALB) = (AMB))
Theorem	sspm 8647	Vector subtraction on a subspace is a restriction of vector subtraction on the parent space.
|- Y = (BaseSet` W)   &   |- M = (-v` U)   &   |- L = (-v` W)   &   |- H = (SubSp` U)   =>   |- ((U e. NrmCVec /\ W e. H) -> L = (M |` (Y X. Y)))
Theorem	sspz 8648	The zero vector of a subspace is the same as the parent's.
|- Z = (0v` U)   &   |- Q = (0v` W)   &   |- H = (SubSp` U)   =>   |- ((U e. NrmCVec /\ W e. H) -> Q = Z)
Theorem	sspn 8649	The norm on a subspace is a restriction of the norm on the parent space.
|- Y = (BaseSet` W)   &   |- N = (norm` U)   &   |- M = (norm` W)   &   |- H = (SubSp` U)   =>   |- ((U e. NrmCVec /\ W e. H) -> M = (N |` Y))
Theorem	sspnval 8650	The norm on a subspace in terms of the norm on the parent space.
|- Y = (BaseSet` W)   &   |- N = (norm` U)   &   |- M = (norm` W)   &   |- H = (SubSp` U)   =>   |- ((U e. NrmCVec /\ W e. H /\ A e. Y) -> (M` A) = (N` A))
Theorem	sspival 8651	The inner product on a subspace in terms of the inner product on the parent space.
|- Y = (BaseSet` W)   &   |- P = (.i` U)   &   |- Q = (.i` W)   &   |- H = (SubSp` U)   =>   |- (((U e. NrmCVec /\ W e. H) /\ (A e. Y /\ B e. Y)) -> (AQB) = (APB))
Theorem	sspi 8652	The inner product on a subspace is a restriction of the inner product on the parent space.
|- Y = (BaseSet` W)   &   |- P = (.i` U)   &   |- Q = (.i` W)   &   |- H = (SubSp` U)   =>   |- ((U e. NrmCVec /\ W e. H) -> Q = (P |` (Y X. Y)))
Theorem	sspimsval 8653	The induced metric on a subspace in terms of the induced metric on the parent space.
|- Y = (BaseSet` W)   &   |- D = (IndMet` U)   &   |- C = (IndMet` W)   &   |- H = (SubSp` U)   =>   |- (((U e. NrmCVec /\ W e. H) /\ (A e. Y /\ B e. Y)) -> (ACB) = (ADB))
Theorem	sspims 8654	The induced metric on a subspace is a restriction of the induced metric on the parent space.
|- Y = (BaseSet` W)   &   |- D = (IndMet` U)   &   |- C = (IndMet` W)   &   |- H = (SubSp` U)   =>   |- ((U e. NrmCVec /\ W e. H) -> C = (D |` (Y X. Y)))
Operators on complex vector spaces
Definitions and basic properties
Syntax	clno 8655	Extend class notation with the class of linear operators on normed complex vector spaces.
class LnOp
Syntax	cnmo 8656	Extend class notation with the class of operator norms on normed complex vector spaces.
class normOp
Syntax	cblo 8657	Extend class notation with the class of bounded linear operators on normed complex vector spaces.
class BLnOp
Syntax	c0o 8658	Extend class notation with the class of zero operators on normed complex vector spaces.
class 0op
Definition	df-lno 8659	Define the class of linear operators between two normed complex vector spaces. In the literature, an operator may be a partial function, i.e. the domain of an operator is not necessarily the entire vector space. However, since the domain of a linear operator is a vector subspace, we define it with a complete function for convenience and will use subset relations to specify the partial function case.
|- LnOp = {<.<.u, w>., o>. | ((u e. NrmCVec /\ w e. NrmCVec) /\ o = {t | (t:(BaseSet` u)-->(BaseSet` w) /\ A.x e. (BaseSet` u)A.y e. CC A.z e. (BaseSet` u)(t` (x(+v` u)(y(.s` u)z))) = ((t` x)(+v` w)(y(.s` w)(t` z))))})}
Definition	df-nmo 8660	Define the norm of an operator between two normed complex vector spaces. This definition produces an operator norm function for each pair of vector spaces <.u, w>.. Based on definition of linear operator norm in [AkhiezerGlazman] p. 39, although we define it for all operators for convenience. It isn't necessarily meaningful for nonlinear operators, since it doesn't take into account operator values at vectors with norm greater than 1. See Equation 2 of [Kreyszig] p. 92 for a definition that does (although it ignores the value at the zero vector). However, operator norms are rarely if ever used for nonlinear operators.
|- normOp = {<.<.u, w>., n>. | ((u e. NrmCVec /\ w e. NrmCVec) /\ n = {<.t, y>. | (t:(BaseSet` u)-->(BaseSet` w) /\ y = sup({x | E.z e. (BaseSet` u)(((norm` u)` z) <_ 1 /\ x = ((norm` w)` (t` z)))}, RR*, < ))})}
Definition	df-blo 8661	Define the class of bounded linear operators between two normed complex vector spaces.
|- BLnOp = {<.<.u, w>., o>. | ((u e. NrmCVec /\ w e. NrmCVec) /\ o = {t e. (u LnOp w) | ((unormOpw)` t) < +oo})}
Definition	df-0o 8662	Define the zero operator between two normed complex vector spaces.
|- 0op = {<.<.u, w>., o>. | ((u e. NrmCVec /\ w e. NrmCVec) /\ o = ((BaseSet` u) X. {(0v` w)}))}
Syntax	caj 8663	Adjoint of an operator.
class adj
Syntax	chmo 8664	Set of Hermitional (self-adjoint) operators.
class HmOp
Definition	df-aj 8665	Define the adjoint of an operator (if it exists). The domain of UadjW is the set of all operators from U to W that have an adjoint. Definition 3.9-1 of [Kreyszig] p. 196, although we don't require that U and W be Hilbert spaces nor that the operators be linear. Although we define it for any normed vector space for convenience, the definition is meaningful only for inner product spaces.
|- adj = {<.<.u, w>., a>. | ((u e. NrmCVec /\ w e. NrmCVec) /\ a = {<.t, s>. | (t:(BaseSet` u)-->(BaseSet` w) /\ s:(BaseSet` w)-->(BaseSet` u) /\ A.x e. (BaseSet` u)A.y e. (BaseSet` w)((t` x)(.i` w)y) = (x(.i` u)(s` y)))})}
Definition	df-hmo 8666	Define the set of Hermitian (self-adjoint) operators on a normed complex vector space (normally a Hilbert space). Although we define it for any normed vector space for convenience, the definition is meaningful only for inner product spaces.
|- HmOp = {<.u, o>. | (u e. NrmCVec /\ o = {t e. dom ( uadju) | ((uadju)` t) = t})}
Theorem	lnoval 8667	The set of linear operators between two normed complex vector spaces.
|- X = (BaseSet` U)   &   |- Y = (BaseSet` W)   &   |- G = (+v` U)   &   |- H = (+v` W)   &   |- R = (.s` U)   &   |- S = (.s` W)   &   |- L = (U LnOp W)   =>   |- ((U e. NrmCVec /\ W e. NrmCVec) -> L = {t | (t:X-->Y /\ A.x e. X A.y e. CC A.z e. X (t` (xG(yRz))) = ((t` x)H(yS(t` z))))})
Theorem	islno 8668	The predicate "is a linear operator."
|- X = (BaseSet` U)   &   |- Y = (BaseSet` W)   &   |- G = (+v` U)   &   |- H = (+v` W)   &   |- R = (.s` U)   &   |- S = (.s` W)   &   |- L = (U LnOp W)   =>   |- ((U e. NrmCVec /\ W e. NrmCVec) -> (T e. L <-> (T:X-->Y /\ A.x e. X A.y e. CC A.z e. X (T` (xG(yRz))) = ((T` x)H(yS(T` z))))))
Theorem	lnolin 8669	Basic linearity property of a linear operator.
|- X = (BaseSet` U)   &   |- Y = (BaseSet` W)   &   |- G = (+v` U)   &   |- H = (+v` W)   &   |- R = (.s` U)   &   |- S = (.s` W)   &   |- L = (U LnOp W)   =>   |- (((U e. NrmCVec /\ W e. NrmCVec /\ T e. L) /\ (A e. X /\ B e. CC /\ C e. X)) -> (T` (AG(BRC))) = ((T` A)H(BS(T` C))))
Theorem	lnof 8670	A linear operator is a mapping.
|- X = (BaseSet` U)   &   |- Y = (BaseSet` W)   &   |- L = (U LnOp W)   =>   |- ((U e. NrmCVec /\ W e. NrmCVec /\ T e. L) -> T:X-->Y)
Theorem	lno0 8671	The value of a linear operator at zero is zero.
|- X = (BaseSet` U)   &   |- Y = (BaseSet` W)   &   |- Q = (0v` U)   &   |- Z = (0v` W)   &   |- L = (U LnOp W)   =>   |- ((U e. NrmCVec /\ W e. NrmCVec /\ T e. L) -> (T` Q) = Z)
Theorem	lnocoi 8672	The composition of two linear operators is linear.
|- L = (U LnOp W)   &   |- M = (W LnOp X)   &   |- N = (U LnOp X)   &   |- U e. NrmCVec   &   |- W e. NrmCVec   &   |- X e. NrmCVec   &   |- S e. L   &   |- T e. M   =>   |- (T o. S) e. N
Theorem	lnoadd 8673	Addition property of a linear operator.
|- X = (BaseSet` U)   &   |- G = (+v` U)   &   |- H = (+v` W)   &   |- L = (U LnOp W)   =>   |- (((U e. NrmCVec /\ W e. NrmCVec /\ T e. L) /\ (A e. X /\ B e. X)) -> (T` (AGB)) = ((T` A)H(T` B)))
Theorem	lnosub 8674	Subtraction property of a linear operator.
|- X = (BaseSet` U)   &   |- M = (-v` U)   &   |- N = (-v` W)   &   |- L = (U LnOp W)   =>   |- (((U e. NrmCVec /\ W e. NrmCVec /\ T e. L) /\ (A e. X /\ B e. X)) -> (T` (AMB)) = ((T` A)N(T` B)))
Theorem	lnomul 8675	Scalar multiplication property of a linear operator.
|- X = (BaseSet` U)   &   |- R = (.s` U)   &   |- S = (.s` W)   &   |- L = (U LnOp W)   =>   |- (((U e. NrmCVec /\ W e. NrmCVec /\ T e. L) /\ (A e. CC /\ B e. X)) -> (T` (ARB)) = (AS(T` B)))
Theorem	nvcnpi3 8676	Epsilon-delta property of a linear operator continuous at a point.)
|- X = (BaseSet` U)   &   |- M = (norm` U)   &   |- N = (norm` W)   &   |- R = (-v` U)   &   |- C = (IndMet` U)   &   |- D = (IndMet` W)   &   |- J = (Open` C)   &   |- K = (Open` D)   &   |- L = (U LnOp W)   &   |- T e. L   =>   |- (((U e. NrmCVec /\ W e. NrmCVec /\ P e. X) /\ (T e. ((J CnP K)` P) /\ A e. RR /\ 0 < A)) -> E.x e. RR (0 < x /\ A.y e. X ((M` (yRP)) <_ x -> (N` (T` (yRP))) <_ A)))
Theorem	nvcnpi4 8677	Epsilon-delta property of a linear operator continuous at a point.)
|- X = (BaseSet` U)   &   |- M = (norm` U)   &   |- N = (norm` W)   &   |- R = (-v` U)   &   |- C = (IndMet` U)   &   |- D = (IndMet` W)   &   |- J = (Open` C)   &   |- K = (Open` D)   &   |- L = (U LnOp W)   &   |- T e. L   =>   |- (((U e. NrmCVec /\ W e. NrmCVec /\ P e. X) /\ (T e. ((J CnP K)` P) /\ A e. RR /\ 0 < A)) -> E.x e. RR (0 < x /\ A.y e. X ((M` (PRy)) <_ x -> (N` (T` (PRy))) <_ A)))
Theorem	nvo00 8678	Two ways to express a zero operator.
|- X = (BaseSet` U)   =>   |- ((U e. NrmCVec /\ T:X-->Y) -> (T = (X X. {Z}) <-> ran T = {Z}))
Theorem	nmofval 8679	The operator norm function.
|- X = (BaseSet` U)   &   |- Y = (BaseSet` W)   &   |- L = (norm` U)   &   |- M = (norm` W)   &   |- N = (UnormOpW)   =>   |- ((U e. NrmCVec /\ W e. NrmCVec) -> N = {<.t, y>. | (t:X-->Y /\ y = sup({x | E.z e. X ((L` z) <_ 1 /\ x = (M` (t` z)))}, RR*, < ))})
Theorem	nmoval 8680	The operator norm function.
|- X = (BaseSet` U)   &   |- Y = (BaseSet` W)   &   |- L = (norm` U)   &   |- M = (norm` W)   &   |- N = (UnormOpW)   =>   |- ((U e. NrmCVec /\ W e. NrmCVec /\ T:X-->Y) -> (N` T) = sup({x | E.z e. X ((L` z) <_ 1 /\ x = (M` (T` z)))}, RR*, < ))
Theorem	nmosetre 8681	The set in the supremum of the operator norm definition df-nmo 8660 is a set of reals.
|- Y = (BaseSet` W)   &   |- N = (norm` W)   =>   |- ((W e. NrmCVec /\ T:X-->Y) -> {x | E.z e. X ((M` z) <_ 1 /\ x = (N` (T` z)))} (_ RR)
Theorem	nmosetn0 8682	The set in the supremum of the operator norm definition df-nmo 8660 is nonempty.
|- X = (BaseSet` U)   &   |- Z = (0v` U)   &   |- M = (norm` U)   =>   |- (U e. NrmCVec -> (N` (T` Z)) e. {x | E.y e. X ((M` y) <_ 1 /\ x = (N` (T` y)))})
Theorem	nmoxr 8683	The norm of an operator is an extended real.
|- X = (BaseSet` U)   &   |- Y = (BaseSet` W)   &   |- N = (UnormOpW)   =>   |- ((U e. NrmCVec /\ W e. NrmCVec /\ T:X-->Y) -> (N` T) e. RR*)
Theorem	nmoge0 8684	The norm of an operator is nonnegative.
|- X = (BaseSet` U)   &   |- Y = (BaseSet` W)   &   |- N = (UnormOpW)   =>   |- ((U e. NrmCVec /\ W e. NrmCVec /\ T:X-->Y) -> 0 <_ (N` T))
Theorem	nmorepnf 8685	The norm of an operator is either real or plus infinity.
|- X = (BaseSet` U)   &   |- Y = (BaseSet` W)   &   |- N = (UnormOpW)   =>   |- ((U e. NrmCVec /\ W e. NrmCVec /\ T:X-->Y) -> ((N` T) e. RR <-> (N` T) =/= +oo))
Theorem	nmoreltpnf 8686	The norm of any operator is real iff it is less than plus infinity.
|- X = (BaseSet` U)   &   |- Y = (BaseSet` W)   &   |- N = (UnormOpW)   =>   |- ((U e. NrmCVec /\ W e. NrmCVec /\ T:X-->Y) -> ((N` T) e. RR <-> (N` T) < +oo))
Theorem	nmogtmnf 8687	The norm of an operator is greater than minus infinity.
|- X = (BaseSet` U)   &   |- Y = (BaseSet` W)   &   |- N = (UnormOpW)   =>   |- ((U e. NrmCVec /\ W e. NrmCVec /\ T:X-->Y) -> -oo < (N` T))
Theorem	nmolb 8688	A lower bound for an operator norm.
|- X = (BaseSet` U)   &   |- Y = (BaseSet` W)   &   |- L = (norm` U)   &   |- M = (norm` W)   &   |- N = (UnormOpW)   =>   |- (((U e. NrmCVec /\ W e. NrmCVec /\ T:X-->Y) /\ (A e. X /\ (L` A) <_ 1)) -> (M` (T` A)) <_ (N` T))
Theorem	nmoubi 8689	An upper bound for an operator norm.
|- X = (BaseSet` U)   &   |- Y = (BaseSet` W)   &   |- L = (norm` U)   &   |- M = (norm` W)   &   |- N = (UnormOpW)   &   |- U e. NrmCVec   &   |- W e. NrmCVec   =>   |- ((T:X-->Y /\ A e. RR* /\ A.x e. X ((L` x) <_ 1 -> (M` (T` x)) <_ A)) -> (N` T) <_ A)
Theorem	nmoub3i 8690	An upper bound for an operator norm.
|- X = (BaseSet` U)   &   |- Y = (BaseSet` W)   &   |- L = (norm` U)   &   |- M = (norm` W)   &   |- N = (UnormOpW)   &   |- U e. NrmCVec   &   |- W e. NrmCVec   =>   |- ((T:X-->Y /\ A e. RR /\ A.x e. X (M` (T` x)) <_ (A x. (L` x))) -> (N` T) <_ (abs` A))
Theorem	nmoub2i 8691	An upper bound for an operator norm.
|- X = (BaseSet` U)   &   |- Y = (BaseSet` W)   &   |- L = (norm` U)   &   |- M = (norm` W)   &   |- N = (UnormOpW)   &   |- U e. NrmCVec   &   |- W e. NrmCVec   =>   |- ((T:X-->Y /\ (A e. RR /\ 0 <_ A) /\ A.x e. X (M` (T` x)) <_ (A x. (L` x))) -> (N` T) <_ A)
Theorem	nmobndi 8692	Two ways to express that an operator is bounded.
|- X = (BaseSet` U)   &   |- Y = (BaseSet` W)   &   |- L = (norm` U)   &   |- M = (norm` W)   &   |- N = (UnormOpW)   &   |- U e. NrmCVec   &   |- W e. NrmCVec   =>   |- (T:X-->Y -> ((N` T) e. RR <-> E.r e. RR A.y e. X ((L` y) <_ 1 -> (M` (T` y)) <_ r)))
Theorem	nmounbi 8693	Two ways two express that an operator is unbounded.
|- X = (BaseSet` U)   &   |- Y = (BaseSet` W)   &   |- L = (norm` U)   &   |- M = (norm` W)   &   |- N = (UnormOpW)   &   |- U e. NrmCVec   &   |- W e. NrmCVec   =>   |- (T:X-->Y -> ((N` T) = +oo <-> A.r e. RR E.y e. X ((L` y) <_ 1 /\ r < (M` (T` y)))))
Theorem	nmounbseqi 8694	An unbounded operator determines an unbounded sequence.
|- X = (BaseSet` U)   &   |- Y = (BaseSet` W)   &   |- L = (norm` U)   &   |- M = (norm` W)   &   |- N = (UnormOpW)   &   |- U e. NrmCVec   &   |- W e. NrmCVec   =>   |- ((T:X-->Y /\ (N` T) = +oo) -> E.f(f:NN-->X /\ A.k e. NN ((L` (f` k)) <_ 1 /\ k < (M` (T` (f` k))))))
Theorem	nmobndseqi 8695	A bounded sequence determines a bounded operator.
|- X = (BaseSet` U)   &   |- Y = (BaseSet` W)   &   |- L = (norm` U)   &   |- M = (norm` W)   &   |- N = (UnormOpW)   &   |- U e. NrmCVec   &   |- W e. NrmCVec   =>   |- ((T:X-->Y /\ A.f((f:NN-->X /\ A.k e. NN (L` (f` k)) <_ 1) -> E.k e. NN (M` (T` (f` k))) <_ k)) -> (N` T) e. RR)
Theorem	bloval 8696	The class of bounded linear operators between two normed complex vector spaces.
|- N = (UnormOpW)   &   |- L = (U LnOp W)   &   |- B = (U BLnOp W)   =>   |- ((U e. NrmCVec /\ W e. NrmCVec) -> B = {t e. L | (N` t) < +oo})
Theorem	isblo 8697	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 8698	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 8699	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 8700	A bounded operator is an operator.
|- X = (BaseSet` U)   &   |- Y = (BaseSet` W)   &   |- B = (U BLnOp W)   =>   |- ((U e. NrmCVec /\ W e. NrmCVec /\ T e. B) -> T:X-->Y)