mmtheorems85 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 8401-8500 - Page 85 of 108

Type	Label	Description
Statement
Theorem	sspmval 8401	Vector addition on a subspace in terms of vector addition on the parent space.
|- Y = (Base` 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 8402	Vector subtraction on a subspace is a restriction of vector subtraction on the parent space.
|- Y = (Base` W)   &   |- M = (-v` U)   &   |- L = (-v` W)   &   |- H = (SubSp` U)   =>   |- ((U e. NrmCVec /\ W e. H) -> L = (M |` (Y X. Y)))
Theorem	sspz 8403	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 8404	The norm on a subspace is a restriction of the norm on the parent space.
|- Y = (Base` W)   &   |- N = (norm` U)   &   |- M = (norm` W)   &   |- H = (SubSp` U)   =>   |- ((U e. NrmCVec /\ W e. H) -> M = (N |` Y))
Theorem	sspnval 8405	The norm on a subspace in terms of the norm on the parent space.
|- Y = (Base` 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 8406	The inner product on a subspace in terms of the inner product on the parent space.
|- Y = (Base` 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 8407	The inner product on a subspace is a restriction of the inner product on the parent space.
|- Y = (Base` W)   &   |- P = (.i` U)   &   |- Q = (.i` W)   &   |- H = (SubSp` U)   =>   |- ((U e. NrmCVec /\ W e. H) -> Q = (P |` (Y X. Y)))
Theorem	sspimsval 8408	The induced metric on a subspace in terms of the induced metric on the parent space.
|- Y = (Base` 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 8409	The induced metric on a subspace is a restriction of the induced metric on the parent space.
|- Y = (Base` 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 8410	Extend class notation with the class of linear operators on normed complex vector spaces.
class LnOp
Syntax	cnmo 8411	Extend class notation with the class of operator norms on normed complex vector spaces.
class normOp
Syntax	cblo 8412	Extend class notation with the class of bounded linear operators on normed complex vector spaces.
class BLnOp
Syntax	c0o 8413	Extend class notation with the class of zero operators on normed complex vector spaces.
class 0op
Definition	df-lno 8414	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:(Base` u)-->(Base` w) /\ A.x e. (Base` u)A.y e. CC A.z e. (Base` u)(t` (x(+v` u)(y(.s` u)z))) = ((t` x)(+v` w)(y(.s` w)(t` z))))})}
Definition	df-nmo 8415	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:(Base` u)-->(Base` w) /\ y = sup({x | E.z e. (Base` u)(((norm` u)` z) <_ 1 /\ x = ((norm` w)` (t` z)))}, RR*, < ))})}
Definition	df-blo 8416	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 8417	Define the zero operator between two normed complex vector spaces.
|- 0op = {<.<.u, w>., o>. | ((u e. NrmCVec /\ w e. NrmCVec) /\ o = ((Base` u) X. {(0v` w)}))}
Syntax	caj 8418	Adjoint of an operator.
class adj
Syntax	chmo 8419	Set of Hermitional (self-adjoint) operators.
class HmOp
Definition	df-aj 8420	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:(Base` u)-->(Base` w) /\ s:(Base` w)-->(Base` u) /\ A.x e. (Base` u)A.y e. (Base` w)((t` x)(.i` w)y) = (x(.i` u)(s` y)))})}
Definition	df-hmo 8421	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 8422	The set of linear operators between two normed complex vector spaces.
|- X = (Base` U)   &   |- Y = (Base` 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 8423	The predicate "is a linear operator."
|- X = (Base` U)   &   |- Y = (Base` 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 8424	Basic linearity property of a linear operator.
|- X = (Base` U)   &   |- Y = (Base` 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 8425	A linear operator is a mapping.
|- X = (Base` U)   &   |- Y = (Base` W)   &   |- L = (U LnOp W)   =>   |- ((U e. NrmCVec /\ W e. NrmCVec /\ T e. L) -> T:X-->Y)
Theorem	lno0 8426	The value of a linear operator at zero is zero.
|- X = (Base` U)   &   |- Y = (Base` 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 8427	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 8428	Addition property of a linear operator.
|- X = (Base` 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 8429	Subtraction property of a linear operator.
|- X = (Base` 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)