mmtheorems116 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 11501-11600 - Page 116 of 175

Type	Label	Description
Statement
Theorem	hmdmadj 11501	Every Hermitian operator has an adjoint.
|- (T e. HrmOp -> T e. dom adjh)
Theorem	hmopadj2 11502	An operator is Hermitian iff it is self-adjoint. Definition of Hermitian in [Halmos] p. 41.
|- (T e. dom adjh -> (T e. HrmOp <-> (adjh` T) = T))
Theorem	hmoplin 11503	A Hermitian operator is linear.
|- (T e. HrmOp -> T e. LinOp)
Theorem	braval 11504	The bra of a vector, expressed as <.A | in Dirac notation. See df-bra 11413.
|- (A e. ~H -> (bra` A) = {<.x, y>. | (x e. ~H /\ y = (x .ih A))})
Theorem	bravalval 11505	A bra-ket juxtaposition, expressed as <.A | B>. in Dirac notation, equals the inner product of the vectors. Based on definition of bra in [Prugovecki] p. 186.
|- ((A e. ~H /\ B e. ~H) -> ((bra` A)` B) = (B .ih A))
Theorem	braadd 11506	Linearity property of bra for addition.
|- ((A e. ~H /\ B e. ~H /\ C e. ~H) -> ((bra` A)` (B +h C)) = (((bra` A)` B) + ((bra` A)` C)))
Theorem	bramul 11507	Linearity property of bra for multiplication.
|- ((A e. ~H /\ B e. CC /\ C e. ~H) -> ((bra` A)` (B .h C)) = (B x. ((bra` A)` C)))
Theorem	brafn 11508	The bra function is a functional.
|- (A e. ~H -> (bra` A):~H-->CC)
Theorem	bralnfn 11509	The Dirac bra function is a linear functional.
|- (A e. ~H -> (bra` A) e. LinFn)
Theorem	bracl 11510	Closure of the bra function.
|- ((A e. ~H /\ B e. ~H) -> ((bra` A)` B) e. CC)
Theorem	bra0 11511	The Dirac bra of the zero vector.
|- (bra` 0h) = (~H X. {0})
Theorem	brafnmul 11512	Anti-linearity property of bra functional for multiplication.
|- ((A e. CC /\ B e. ~H) -> (bra` (A .h B)) = ((*` A) .fn (bra` B)))
Theorem	kbval 11513	The outer product of two vectors, expressed as | A>. <.B | in Dirac notation. See df-kb 11414.
|- ((A e. ~H /\ B e. ~H) -> (A ketbra B) = {<.x, y>. | (x e. ~H /\ y = ((x .ih B) .h A))})
Theorem	kbop 11514	The outer product of two vectors, expressed as | A>. <.B | in Dirac notation, is an operator.
|- ((A e. ~H /\ B e. ~H) -> (A ketbra B):~H-->~H)
Theorem	kbvalval 11515	The value of the operator resulting from the outer product | A>. <.B | of two vectors. Equation 8.1 of [Prugovecki] p. 376.
|- ((A e. ~H /\ B e. ~H /\ C e. ~H) -> ((A ketbra B)` C) = ((C .ih B) .h A))
Theorem	kbmul 11516	Multiplication property of outer product.
|- ((A e. CC /\ B e. ~H /\ C e. ~H) -> ((A .h B) ketbra C) = (B ketbra ((*` A) .h C)))
Theorem	kbpj 11517	If a vector A has norm 1, the outer product | A>. <.A | is the projector onto the subspace spanned by A. http://en.wikipedia.org/wiki/Bra-ket#Linear%5Foperators.
|- ((A e. ~H /\ (normh` A) = 1) -> (A ketbra A) = (proj` (span` {A})))
Theorem	eleigvec 11518	Membership in the set of eigenvectors of a Hilbert space operator.
|- (T:~H-->~H -> (A e. (eigvec` T) <-> (A e. ~H /\ A =/= 0h /\ E.x e. CC (T` A) = (x .h A))))
Theorem	eleigvec2 11519	Membership in the set of eigenvectors of a Hilbert space operator.
|- (T:~H-->~H -> (A e. (eigvec` T) <-> (A e. ~H /\ A =/= 0h /\ (T` A) e. (span` {A}))))
Theorem	eleigveccl 11520	Closure of an eigenvector of a Hilbert space operator.
|- ((T:~H-->~H /\ A e. (eigvec` T)) -> A e. ~H)
Theorem	eigval1 11521	The eigenvalue of an eigenvector of a Hilbert space operator.
|- ((T:~H-->~H /\ A e. (eigvec` T)) -> ((eigval` T)` A) = (((T` A) .ih A) / ((normh` A)^2)))
Theorem	eigvalcl 11522	An eigenvalue is a complex number.
|- ((T:~H-->~H /\ A e. (eigvec` T)) -> ((eigval` T)` A) e. CC)
Theorem	eigvec1 11523	Property of an eigenvector.
|- ((T:~H-->~H /\ A e. (eigvec` T)) -> ((T` A) = (((eigval` T)` A) .h A) /\ A =/= 0h))
Theorem	eighmre 11524	The eigenvalues of a Hermitian operator are real. Equation 1.30 of [Hughes] p. 49.
|- ((T e. HrmOp /\ A e. (eigvec` T)) -> ((eigval` T)` A) e. RR)
Theorem	eighmorth 11525	Eigenvectors of a Hermitian operator with distinct eigenvalues are orthogonal. Equation 1.31 of [Hughes] p. 49.
|- (((T e. HrmOp /\ A e. (eigvec` T)) /\ (B e. (eigvec` T) /\ ((eigval` T)` A) =/= ((eigval` T)` B))) -> (A .ih B) = 0)
Theorem	nmopnegi 11526	Value of the norm of the negative of a Hilbert space operator. Unlike nmophmi 11598, the operator does not have to be bounded.
|- T:~H-->~H   =>   |- (normop` (-u1 .op T)) = (normop` T)
Theorem	lnop0 11527	The value of a linear Hilbert space operator at zero is zero. Remark in [Beran] p. 99.
|- (T e. LinOp -> (T` 0h) = 0h)
Theorem	lnopmul 11528	Multiplicative property of a linear Hilbert space operator.
|- ((T e. LinOp /\ A e. CC /\ B e. ~H) -> (T` (A .h B)) = (A .h (T` B)))
Theorem	lnopli 11529	Basic scalar product property of a linear Hilbert space operator.
|- T e. LinOp   =>   |- ((A e. CC /\ B e. ~H /\ C e. ~H) -> (T` ((A .h B) +h C)) = ((A .h (T` B)) +h (T` C)))
Theorem	lnopfi 11530	A linear Hilbert space operator is a Hilbert space operator.
|- T e. LinOp   =>   |- T:~H-->~H
Theorem	lnop0i 11531	The value of a linear Hilbert space operator at zero is zero. Remark in [Beran] p. 99.
|- T e. LinOp   =>   |- (T` 0h) = 0h
Theorem	lnopaddi 11532	Additive property of a linear Hilbert space operator.
|- T e. LinOp   =>   |- ((A e. ~H /\ B e. ~H) -> (T` (A +h B)) = ((T` A) +h (T` B)))
Theorem	lnopmuli 11533	Multiplicative property of a linear Hilbert space operator.
|- T e. LinOp   =>   |- ((A e. CC /\ B e. ~H) -> (T` (A .h B)) = (A .h (T` B)))
Theorem	lnopaddmuli 11534	Sum/product property of a linear Hilbert space operator.
|- T e. LinOp   =>   |- ((A e. CC /\ B e. ~H /\ C e. ~H) -> (T` (B +h (A .h C))) = ((T` B) +h (A .h (T` C))))
Theorem	lnopsubi 11535	Subtraction property for a linear Hilbert space operator.
|- T e. LinOp   =>   |- ((A e. ~H /\ B e. ~H) -> (T` (A -h B)) = ((T` A) -h (T` B)))
Theorem	lnopsubmuli 11536	Subtraction/product property of a linear Hilbert space operator.
|- T e. LinOp   =>   |- ((A e. CC /\ B e. ~H /\ C e. ~H) -> (T` (B -h (A .h C))) = ((T` B) -h (A .h (T` C))))
Theorem	lnopmulsubi 11537	Product/subtraction property of a linear Hilbert space operator.
|- T e. LinOp   =>   |- ((A e. CC /\ B e. ~H /\ C e. ~H) -> (T` ((A .h B) -h C)) = ((A .h (T` B)) -h (T` C)))
Theorem	homco2 11538	Move a scalar product out of a composition of operators. The operator T must be linear, unlike homco1 11364 that works for any operators.
|- ((A e. CC /\ T e. LinOp /\ U:~H-->~H) -> (T o. (A .op U)) = (A .op (T o. U)))
Theorem	idunop 11539	The identity function (restricted to Hilbert space) is a unitary operator.
|- ( _I |` ~H) e. UniOp
Theorem	0cnop 11540	The identically zero function is a continuous Hilbert space operator.
|- 0hop e. ConOp
Theorem	0cnfn 11541	The identically zero function is a continuous Hilbert space functional.
|- (~H X. {0}) e. ConFn
Theorem	idcnop 11542	The identity function (restricted to Hilbert space) is a continuous operator.
|- ( _I |` ~H) e. ConOp
Theorem	idhmop 11543	The Hilbert space identity operator is a Hermitian operator.
|- Iop e. HrmOp
Theorem	0hmop 11544	The identically zero function is a Hermitian operator.
|- 0hop e. HrmOp
Theorem	0lnop 11545	The identically zero function is a linear Hilbert space operator.
|- 0hop e. LinOp
Theorem	0lnfn 11546	The identically zero function is a linear Hilbert space functional.
|- (~H X. {0}) e. LinFn
Theorem	nmop0 11547	The norm of the zero operator is zero.
|- (normop` 0hop) = 0
Theorem	nmfn0 11548	The norm of the identically zero functional is zero.
|- (normfn` (~H X. {0})) = 0
Theorem	hmopbdopiHIL 11549	A Hermitian operator is a bounded linear operator (Hellinger-Toeplitz Theorem).
|- T e. HrmOp   =>   |- T e. BndLinOp
Theorem	hmopbdoptHIL 11550	A Hermitian operator is a bounded linear operator (Hellinger-Toeplitz Theorem).
|- (T e. HrmOp -> T e. BndLinOp)
Theorem	hoddii 11551	Distributive law for Hilbert space operator difference. (Interestingly, the reverse distributive law hocsubdiri 11343 does not require linearity.)
|- R e. LinOp   &   |- S:~H-->~H   &   |- T:~H-->~H   =>   |- (R o. (S -op T)) = ((R o. S) -op (R o. T))
Theorem	hoddi 11552	Distributive law for Hilbert space operator difference. (Interestingly, the reverse distributive law hocsubdiri 11343 does not require linearity.)
|- ((R e. LinOp /\ S:~H-->~H /\ T:~H-->~H) -> (R o. (S -op T)) = ((R o. S) -op (R o. T)))
Theorem	nmop0h 11553	The norm of any operator on the trivial Hilbert space is zero. (This is the reason we need ~H =/= 0H in nmopun 11576.)
|- ((~H = 0H /\ T:~H-->~H) -> (normop` T) = 0)
Theorem	idlnop 11554	The identity function (restricted to Hilbert space) is a linear operator.
|- ( _I |` ~H) e. LinOp
Theorem	0bdop 11555	The identically zero operator is bounded.
|- 0hop e. BndLinOp
Theorem	adj0 11556	Adjoint of the zero operator.
|- (adjh` 0hop) = 0hop
Theorem	nmlnop0iALT 11557	A linear operator with a zero norm is identically zero.
|- T e. LinOp   =>   |- ((normop` T) = 0 <-> T = 0hop)
Theorem	nmlnop0iHIL 11558	A linear operator with a zero norm is identically zero.
|- T e. LinOp   =>   |- ((normop` T) = 0 <-> T = 0hop)
Theorem	nmlnopgt0i 11559	A linear Hilbert space operator that is not identically zero has a positive norm.
|- T e. LinOp   =>   |- (T =/= 0hop <-> 0 < (normop` T))
Theorem	nmlnop0 11560	A linear operator with a zero norm is identically zero.
|- (T e. LinOp -> ((normop` T) = 0 <-> T = 0hop))
Theorem	nmlnopne0 11561	A linear operator with a nonzero norm is nonzero.
|- (T e. LinOp -> ((normop` T) =/= 0 <-> T =/= 0hop))
Theorem	lnopmi 11562	The scalar product of a linear operator is a linear operator.
|- T e. LinOp   =>   |- (A e. CC -> (A .op T) e. LinOp)
Theorem	lnophsi 11563	The sum of two linear operators is linear.
|- S e. LinOp   &   |- T e. LinOp   =>   |- (S +op T) e. LinOp
Theorem	lnophdi 11564	The difference of two linear operators is linear.
|- S e. LinOp   &   |- T e. LinOp   =>   |- (S -op T) e. LinOp
Theorem	lnopcoi 11565	The composition of two linear operators is linear.
|- S e. LinOp   &   |- T e. LinOp   =>   |- (S o. T) e. LinOp
Theorem	lnopco0i 11566	The composition of a linear operator with one whose norm is zero.
|- S e. LinOp   &   |- T e. LinOp   =>   |- ((normop` T) = 0 -> (normop` (S o. T)) = 0)
Theorem	lnopeq0lem1 11567	Lemma for lnopeq0i 11569. Apply the generalized polarization identity polid2i 10657 to the quadratic form ((T` x), x).
Theorem	lnopeq0lem2 11568	Lemma for lnopeq0i 11569.
Theorem	lnopeq0i 11569	A condition implying that a linear Hilbert space operator is identically zero. Unlike ho01i 11391 for arbitrary operators, when the operator is linear we need to consider only the values of the quadratic form (T` x) .ih x).
|- T e. LinOp   =>   |- (A.x e. ~H ((T` x) .ih x) = 0 <-> T = 0hop)
Theorem	lnopeqi 11570	Two linear Hilbert space operators are equal iff their quadratic forms are equal.
|- T e. LinOp   &   |- U e. LinOp   =>   |- (A.x e. ~H ((T` x) .ih x) = ((U` x) .ih x) <-> T = U)
Theorem	lnopeq 11571	Two linear Hilbert space operators are equal iff their quadratic forms are equal.
|- ((T e. LinOp /\ U e. LinOp) -> (A.x e. ~H ((T` x) .ih x) = ((U` x) .ih x) <-> T = U))
Theorem	lnopunilem1 11572	Lemma for lnopunii 11574.
Theorem	lnopunilem2 11573	Lemma for lnopunii 11574.
Theorem	lnopunii 11574	If a linear operator (whose range is ~H) is idempotent in the norm, the operator is unitary. Similar to theorem in [AkhiezerGlazman] p. 73.
|- T e. LinOp   &   |- T:~H-onto->~H   &   |- A.x e. ~H (normh` (T` x)) = (normh` x)   =>   |- T e. UniOp
Theorem	elunop2 11575	An operator is unitary iff it is linear, onto, and idempotent in the norm. Similar to theorem in [AkhiezerGlazman] p. 73, and its converse.
|- (T e. UniOp <-> (T e. LinOp /\ T:~H-onto->~H /\ A.x e. ~H (normh` (T` x)) = (normh` x)))
Theorem	nmopun 11576	Norm of a unitary Hilbert space operator.
|- ((~H =/= 0H /\ T e. UniOp) -> (normop` T) = 1)
Theorem	unopbd 11577	A unitary operator is a bounded linear operator.
|- (T e. UniOp -> T e. BndLinOp)
Theorem	lnophmlem1 11578	Lemma for lnophmi 11580.
Theorem	lnophmlem2 11579	Lemma for lnophmi 11580. Warning: The HTML proof page is 1/2 megabyte in size.
Theorem	lnophmi 11580	A linear operator is Hermitian if x .ih (T` x) takes only real values. Remark in [ReedSimon] p. 195.
|- T e. LinOp   &   |- A.x e. ~H (x .ih (T` x)) e. RR   =>   |- T e. HrmOp
Theorem	lnophm 11581	A linear operator is Hermitian if x .ih (T` x) takes only real values. Remark in [ReedSimon] p. 195.
|- ((T e. LinOp /\ A.x e. ~H (x .ih (T` x)) e. RR) -> T e. HrmOp)
Theorem	hmops 11582	The sum of two Hermitian operators is Hermitian.
|- ((T e. HrmOp /\ U e. HrmOp) -> (T +op U) e. HrmOp)
Theorem	hmopm 11583	The scalar product of a Hermitian operator with a real is Hermitian.
|- ((A e. RR /\ T e. HrmOp) -> (A .op T) e. HrmOp)
Theorem	hmopd 11584	The difference of two Hermitian operators is Hermitian.
|- ((T e. HrmOp /\ U e. HrmOp) -> (T -op U) e. HrmOp)
Theorem	hmopco 11585	The composition of two commuting Hermitian operators is Hermitian.
|- ((T e. HrmOp /\ U e. HrmOp /\ (T o. U) = (U o. T)) -> (T o. U) e. HrmOp)
Theorem	nmbdoplbi 11586	A lower bound for the norm of a bounded linear operator.
|- T e. BndLinOp   =>   |- (A e. ~H -> (normh` (T` A)) <_ ((normop` T) x. (normh` A)))
Theorem	nmbdoplb 11587	A lower bound for the norm of a bounded linear Hilbert space operator.
|- ((T e. BndLinOp /\ A e. ~H) -> (normh` (T` A)) <_ ((normop` T) x. (normh` A)))
Theorem	nmcopexlem1 11588	Lemma for nmcopexi 11594 (Theorem 3.5(i) of [Beran] p. 99). A sufficient condition for the norm of an operator to be real, based on its definition and the properties of supremum. Compared to Beran, we use a direct proof instead of a proof by contradiction.
Theorem	nmcopexlem2 11589	Lemma for nmcopexi 11594. Apply definition of continuity. Note that we use 1 instead of 0.5 that Beran uses for epsilon (e = 0.5 in his proof).
Theorem	nmcopexlem3 11590	Lemma for nmcopexi 11594. Move 1 / n out of the norm, using linearity.
Theorem	nmcopexlem4 11591	Lemma for nmcopexi 11594. Properties of the infimum of a collection of integers whose reciprocals are less than a real number y (which will later become the "epsilon" of the epsilon/delta continuity definition df-cnop 11403). Note that `' < in the fourth hypothesis signifies infimum. (This lemma involves only real numbers and is independent of Hilbert space. The first two hypotheses aren't used.)
Theorem	nmcopexlem5 11592	Lemma for nmcopexi 11594.
Theorem	nmcopexlem6 11593	Lemma for nmcopexi 11594. Combine lemmas to obtain the result (with hypotheses to be eliminated).
Theorem	nmcopexi 11594	The norm of a continuous linear Hilbert space operator exists. Theorem 3.5(i) of [Beran] p. 99.
|- T e. LinOp   &   |- T e. ConOp   =>   |- (normop` T) e. RR
Theorem	nmcoplbi 11595	A lower bound for the norm of a continuous linear operator. Theorem 3.5(ii) of [Beran] p. 99.
|- T e. LinOp   &   |- T e. ConOp   =>   |- (A e. ~H -> (normh` (T` A)) <_ ((normop` T) x. (normh` A)))
Theorem	nmcopex 11596	The norm of a continuous linear Hilbert space operator exists. Theorem 3.5(i) of [Beran] p. 99.
|- ((T e. LinOp /\ T e. ConOp) -> (normop` T) e. RR)
Theorem	nmcoplb 11597	A lower bound for the norm of a continuous linear Hilbert space operator. Theorem 3.5(ii) of [Beran] p. 99.
|- ((T e. LinOp /\ T e. ConOp /\ A e. ~H) -> (normh` (T` A)) <_ ((normop` T) x. (normh` A)))
Theorem	nmophmi 11598	The norm of the scalar product of a bounded linear operator.
|- T e. BndLinOp   =>   |- (A e. CC -> (normop` (A .op T)) = ((abs` A) x. (normop` T)))
Theorem	bdophmi 11599	The scalar product of a bounded linear operator is a bounded linear operator.
|- T e. BndLinOp   =>   |- (A e. CC -> (A .op T) e. BndLinOp)
Theorem	lnopconi 11600	A condition equivalent to "T is continuous" when T is linear. Theorem 3.5(iii) of [Beran] p. 99.
|- T e. LinOp   =>   |- (T e. ConOp <-> E.x e. RR A.y e. ~H (normh` (T` y)) <_ (x x. (normh` y)))