mmtheorems117 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 11601-11700 - Page 117 of 175

Type	Label	Description
Statement
Theorem	lnopcon 11601	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))))
Theorem	lnopcnbd 11602	A linear operator is continuous iff it is bounded.
|- (T e. LinOp -> (T e. ConOp <-> T e. BndLinOp))
Theorem	lncnopbd 11603	A continuous linear operator is a bounded linear operator. This theorem justifies our use of "bounded linear" as an interchangeable condition for "continuous linear" used in some textbook proofs.
|- (T e. (LinOp i^i ConOp) <-> T e. BndLinOp)
Theorem	lncnbd 11604	A continuous linear operator is a bounded linear operator.
|- (LinOp i^i ConOp) = BndLinOp
Theorem	lnopcnre 11605	A linear operator is continuous iff it is bounded.
|- (T e. LinOp -> (T e. ConOp <-> (normop` T) e. RR))
Theorem	lnfnli 11606	Basic property of a linear Hilbert space functional.
|- T e. LinFn   =>   |- ((A e. CC /\ B e. ~H /\ C e. ~H) -> (T` ((A .h B) +h C)) = ((A x. (T` B)) + (T` C)))
Theorem	lnfnfi 11607	A linear Hilbert space functional is a functional.
|- T e. LinFn   =>   |- T:~H-->CC
Theorem	lnfn0i 11608	The value of a linear Hilbert space functional at zero is zero. Remark in [Beran] p. 99.
|- T e. LinFn   =>   |- (T` 0h) = 0
Theorem	lnfnaddi 11609	Additive property of a linear Hilbert space functional.
|- T e. LinFn   =>   |- ((A e. ~H /\ B e. ~H) -> (T` (A +h B)) = ((T` A) + (T` B)))
Theorem	lnfnmuli 11610	Multiplicative property of a linear Hilbert space functional.
|- T e. LinFn   =>   |- ((A e. CC /\ B e. ~H) -> (T` (A .h B)) = (A x. (T` B)))
Theorem	lnfnaddmuli 11611	Sum/product property of a linear Hilbert space functional.
|- T e. LinFn   =>   |- ((A e. CC /\ B e. ~H /\ C e. ~H) -> (T` (B +h (A .h C))) = ((T` B) + (A x. (T` C))))
Theorem	lnfnsubi 11612	Subtraction property for a linear Hilbert space functional.
|- T e. LinFn   =>   |- ((A e. ~H /\ B e. ~H) -> (T` (A -h B)) = ((T` A) - (T` B)))
Theorem	lnfn0 11613	The value of a linear Hilbert space functional at zero is zero. Remark in [Beran] p. 99.
|- (T e. LinFn -> (T` 0h) = 0)
Theorem	lnfnmul 11614	Multiplicative property of a linear Hilbert space functional.
|- ((T e. LinFn /\ A e. CC /\ B e. ~H) -> (T` (A .h B)) = (A x. (T` B)))
Theorem	nmbdfnlbi 11615	A lower bound for the norm of a bounded linear functional.
|- (T e. LinFn /\ (normfn` T) e. RR)   =>   |- (A e. ~H -> (abs` (T` A)) <_ ((normfn` T) x. (normh` A)))
Theorem	nmbdfnlb 11616	A lower bound for the norm of a bounded linear functional.
|- ((T e. LinFn /\ (normfn` T) e. RR /\ A e. ~H) -> (abs` (T` A)) <_ ((normfn` T) x. (normh` A)))
Theorem	nmcfnexlem1 11617	Lemma for nmcfnexi 11623. Show a condition for the norm of a functional to exist, based on its definition and the properties of supremum. Compared to Beran, we use a direct proof instead of a proof by contradiction.
Theorem	nmcfnexlem2 11618	Lemma for nmcfnexi 11623. 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	nmcfnexlem3 11619	Lemma for nmcfnexi 11623. Move 1 / n out of the norm, using linearity.
Theorem	nmcfnexlem4 11620	Lemma for nmcfnexi 11623. Properties of the infimum of the collection of integers whose reciprocals are less than the delta of the continuity definition.
Theorem	nmcfnexlem5 11621	Lemma for nmcfnexi 11623.
Theorem	nmcfnexlem6 11622	Lemma for nmcfnexi 11623. Combine lemmas to obtain the result (with hypotheses to be eliminated).
Theorem	nmcfnexi 11623	The norm of a continuous linear Hilbert space functional exists. Theorem 3.5(i) of [Beran] p. 99.
|- T e. LinFn   &   |- T e. ConFn   =>   |- (normfn` T) e. RR
Theorem	nmcfnlbi 11624	A lower bound for the norm of a continuous linear functional. Theorem 3.5(ii) of [Beran] p. 99.
|- T e. LinFn   &   |- T e. ConFn   =>   |- (A e. ~H -> (abs` (T` A)) <_ ((normfn` T) x. (normh` A)))
Theorem	nmcfnex 11625	The norm of a continuous linear Hilbert space functional exists. Theorem 3.5(i) of [Beran] p. 99.
|- ((T e. LinFn /\ T e. ConFn) -> (normfn` T) e. RR)
Theorem	nmcfnlb 11626	A lower bound of the norm of a continuous linear Hilbert space functional. Theorem 3.5(ii) of [Beran] p. 99.
|- ((T e. LinFn /\ T e. ConFn /\ A e. ~H) -> (abs` (T` A)) <_ ((normfn` T) x. (normh` A)))
Theorem	lnfnconi 11627	A condition equivalent to "T is continuous" when T is linear. Theorem 3.5(iii) of [Beran] p. 99.
|- T e. LinFn   =>   |- (T e. ConFn <-> E.x e. RR A.y e. ~H (abs` (T` y)) <_ (x x. (normh` y)))
Theorem	lnfncon 11628	A condition equivalent to "T is continuous" when T is linear. Theorem 3.5(iii) of [Beran] p. 99.
|- (T e. LinFn -> (T e. ConFn <-> E.x e. RR A.y e. ~H (abs` (T` y)) <_ (x x. (normh` y))))
Theorem	lnfncnbd 11629	A linear functional is continuous iff it is bounded.
|- (T e. LinFn -> (T e. ConFn <-> (normfn` T) e. RR))
Theorem	nlelshi 11630	The null space of a linear functional is a subspace.
|- T e. LinFn   =>   |- (null` T) e. SH
Theorem	nlelchi 11631	The null space of a continuous linear functional is a closed subspace. Remark 3.8 of [Beran] p. 103.
|- T e. LinFn   &   |- T e. ConFn   =>   |- (null` T) e. CH
Riesz lemma
Theorem	riesz3i 11632	A continuous linear functional can be expressed as an inner product. Existence part of Theorem 3.9 of [Beran] p. 104.
|- T e. LinFn   &   |- T e. ConFn   =>   |- E.w e. ~H A.v e. ~H (T` v) = (v .ih w)
Theorem	riesz4i 11633	A continuous linear functional can be expressed as an inner product. Uniqueness part of Theorem 3.9 of [Beran] p. 104.
|- T e. LinFn   &   |- T e. ConFn   =>   |- E!w e. ~H A.v e. ~H (T` v) = (v .ih w)
Theorem	riesz4 11634	A continuous linear functional can be expressed as an inner product. Uniqueness part of Theorem 3.9 of [Beran] p. 104. See riesz2 11636 for the bounded linear functional version.
|- (T e. (LinFn i^i ConFn) -> E!w e. ~H A.v e. ~H (T` v) = (v .ih w))
Theorem	riesz1 11635	Part 1 of the Riesz representation theorem for bounded linear functionals. A linear functional is bounded iff its value can be expressed as an inner product. Part of Theorem 17.3 of [Halmos] p. 31. For part 2, see riesz2 11636. For the continuous linear functional version, see riesz3i 11632 and riesz4 11634.
|- (T e. LinFn -> ((normfn` T) e. RR <-> E.y e. ~H A.x e. ~H (T` x) = (x .ih y)))
Theorem	riesz2 11636	Part 2 of the Riesz representation theorem for bounded linear functionals. The value of a bounded linear functional corresponds to a unique inner product. Part of Theorem 17.3 of [Halmos] p. 31. For part 1, see riesz1 11635.
|- ((T e. LinFn /\ (normfn` T) e. RR) -> E!y e. ~H A.x e. ~H (T` x) = (x .ih y))
Adjoints (cont.)
Theorem	cnlnadjlem1 11637	Lemma for cnlnadji 11646 (Theorem 3.10 of [Beran] p. 104: every continuous linear operator has an adjoint). The value of the auxiliary functional G.
Theorem	cnlnadjlem2 11638	Lemma for cnlnadji 11646. G is a continuous linear functional.
Theorem	cnlnadjlem3 11639	Lemma for cnlnadji 11646. By riesz4 11634, B is the unique vector such that (T` v) .ih y) = (v .ih w) for all v.
Theorem	cnlnadjlem4 11640	Lemma for cnlnadji 11646. The values of auxiliary function F are vectors.
Theorem	cnlnadjlem5 11641	Lemma for cnlnadji 11646. F is an adjoint of T (later, we will show it is unique).
Theorem	cnlnadjlem6 11642	Lemma for cnlnadji 11646. F is linear.
Theorem	cnlnadjlem7 11643	Lemma for cnlnadji 11646. Helper lemma to show that F is continuous.
Theorem	cnlnadjlem8 11644	Lemma for cnlnadji 11646. F is continuous.
Theorem	cnlnadjlem9 11645	Lemma for cnlnadji 11646. F provides an example showing the existence of a continuous linear adjoint.
Theorem	cnlnadji 11646	Every continuous linear operator has an adjoint. Theorem 3.10 of [Beran] p. 104.
|- T e. LinOp   &   |- T e. ConOp   =>   |- E.t e. (LinOp i^i ConOp)A.x e. ~H A.y e. ~H ((T` x) .ih y) = (x .ih (t` y))
Theorem	cnlnadjeui 11647	Every continuous linear operator has a unique adjoint. Theorem 3.10 of [Beran] p. 104.
|- T e. LinOp   &   |- T e. ConOp   =>   |- E!t e. (LinOp i^i ConOp)A.x e. ~H A.y e. ~H ((T` x) .ih y) = (x .ih (t` y))
Theorem	cnlnadjeu 11648	Every continuous linear operator has a unique adjoint. Theorem 3.10 of [Beran] p. 104.
|- (T e. (LinOp i^i ConOp) -> E!t e. (LinOp i^i ConOp)A.x e. ~H A.y e. ~H ((T` x) .ih y) = (x .ih (t` y)))
Theorem	cnlnadj 11649	Every continuous linear operator has an adjoint. Theorem 3.10 of [Beran] p. 104.
|- (T e. (LinOp i^i ConOp) -> E.t e. (LinOp i^i ConOp)A.x e. ~H A.y e. ~H ((T` x) .ih y) = (x .ih (t` y)))
Theorem	cnlnssadj 11650	Every continuous linear Hilbert space operator has an adjoint.
|- (LinOp i^i ConOp) C_ dom adjh
Theorem	bdopssadj 11651	Every bounded linear Hilbert space operator has an adjoint.
|- BndLinOp C_ dom adjh
Theorem	bdopadj 11652	Every bounded linear Hilbert space operator has an adjoint.
|- (T e. BndLinOp -> T e. dom adjh)
Theorem	adjbdln 11653	The adjoint of a bounded linear operator is a bounded linear operator.
|- (T e. BndLinOp -> (adjh` T) e. BndLinOp)
Theorem	adjbdlnb 11654	An operator is bounded and linear iff its adjoint is.
|- (T e. BndLinOp <-> (adjh` T) e. BndLinOp)
Theorem	adjbd1o 11655	The mapping of adjoints of bounded linear operators is one-to-one onto.
|- (adjh |` BndLinOp):BndLinOp-1-1-onto->BndLinOp
Theorem	adjlnop 11656	The adjoint of an operator is linear. Proposition 1 of [AkhiezerGlazman] p. 80.
|- (T e. dom adjh -> (adjh` T) e. LinOp)
Theorem	adjsslnop 11657	Every operator with an adjoint is linear.
|- dom adjh C_ LinOp
Theorem	nmopadjlei 11658	Property of the norm of an adjoint. Part of proof of Theorem 3.10 of [Beran] p. 104.
|- T e. BndLinOp   =>   |- (A e. ~H -> (normh` ((adjh` T)` A)) <_ ((normop` T) x. (normh` A)))
Theorem	nmopadjlem 11659	Lemma for nmopadji 11660.
Theorem	nmopadji 11660	Property of the norm of an adjoint. Theorem 3.11(v) of [Beran] p. 106.
|- T e. BndLinOp   =>   |- (normop` (adjh` T)) = (normop` T)
Theorem	adjeq0 11661	An operator is zero iff its adjoint is zero. Theorem 3.11(i) of [Beran] p. 106.
|- (T = 0hop <-> (adjh` T) = 0hop)
Theorem	adjmul 11662	The adjoint of the scalar product of an operator. Theorem 3.11(ii) of [Beran] p. 106.
|- ((A e. CC /\ T e. dom adjh) -> (adjh` (A .op T)) = ((*` A) .op (adjh` T)))
Theorem	adjadd 11663	The adjoint of the sum of two operators. Theorem 3.11(iii) of [Beran] p. 106.
|- ((S e. dom adjh /\ T e. dom adjh) -> (adjh` (S +op T)) = ((adjh` S) +op (adjh` T)))
Theorem	nmoptrii 11664	Triangle inequality for the norms of bounded linear operators.
|- S e. BndLinOp   &   |- T e. BndLinOp   =>   |- (normop` (S +op T)) <_ ((normop` S) + (normop` T))
Theorem	nmopcoi 11665	Upper bound for the norm of the composition of two bounded linear operators.
|- S e. BndLinOp   &   |- T e. BndLinOp   =>   |- (normop` (S o. T)) <_ ((normop` S) x. (normop` T))
Theorem	bdophsi 11666	The sum of two bounded linear operators is a bounded linear operator.
|- S e. BndLinOp   &   |- T e. BndLinOp   =>   |- (S +op T) e. BndLinOp
Theorem	bdophdi 11667	The difference between two bounded linear operators is bounded.
|- S e. BndLinOp   &   |- T e. BndLinOp   =>   |- (S -op T) e. BndLinOp
Theorem	bdopcoi 11668	The composition of two bounded linear operators is bounded.
|- S e. BndLinOp   &   |- T e. BndLinOp   =>   |- (S o. T) e. BndLinOp
Theorem	nmoptri2i 11669	Triangle-type inequality for the norms of bounded linear operators.
|- S e. BndLinOp   &   |- T e. BndLinOp   =>   |- ((normop` S) - (normop` T)) <_ (normop` (S +op T))
Theorem	adjcoi 11670	The adjoint of a composition of bounded linear operators. Theorem 3.11(viii) of [Beran] p. 106.
|- S e. BndLinOp   &   |- T e. BndLinOp   =>   |- (adjh` (S o. T)) = ((adjh` T) o. (adjh` S))
Theorem	nmopcoadji 11671	The norm of an operator composed with its adjoint. Part of Theorem 3.11(vi) of [Beran] p. 106.
|- T e. BndLinOp   =>   |- (normop` ((adjh` T) o. T)) = ((normop` T)^2)
Theorem	nmopcoadj2i 11672	The norm of an operator composed with its adjoint. Part of Theorem 3.11(vi) of [Beran] p. 106.
|- T e. BndLinOp   =>   |- (normop` (T o. (adjh` T))) = ((normop` T)^2)
Theorem	nmopcoadj0i 11673	An operator composed with its adjoint is zero iff the operator is zero. Theorem 3.11(vii) of [Beran] p. 106.
|- T e. BndLinOp   =>   |- ((T o. (adjh` T)) = 0hop <-> T = 0hop)
Quantum computation error bound theorem
Theorem	unierri 11674	If we approximate a chain of unitary transformations (quantum computer gates) F, G by other unitary transformations S, T, the error increases at most additively. Equation 4.73 of [NielsenChuang] p. 195.
|- F e. UniOp   &   |- G e. UniOp   &   |- S e. UniOp   &   |- T e. UniOp   =>   |- (normop` ((F o. G) -op (S o. T))) <_ ((normop` (F -op S)) + (normop` (G -op T)))
Dirac bra-ket notation (cont.)
Theorem	branmfn 11675	The norm of the bra function.
|- (A e. ~H -> (normfn` (bra` A)) = (normh` A))
Theorem	branmfnOLD 11676	The norm of the bra function.
|- (A e. ~H -> (normfn` (bra` A)) = (normh` A))
Theorem	brabn 11677	The bra of a vector is a bounded functional.
|- (A e. ~H -> (normfn` (bra` A)) e. RR)
Theorem	rnbra 11678	The set of bras equals the set of continuous linear functionals.
|- ran bra = (LinFn i^i ConFn)
Theorem	bra11 11679	The bra function maps vectors one-to-one onto the set of continuous linear functionals.
|- bra:~H-1-1-onto->(LinFn i^i ConFn)
Theorem	bracnln 11680	A bra is a continuous linear functional.
|- (A e. ~H -> (bra` A) e. (LinFn i^i ConFn))
Theorem	cnvbraval 11681	Value of the converse of the bra function. Based on the Riesz Lemma riesz4 11634, this very important theorem not only justifies the Dirac bra-ket notation, but allows us to extract a unique vector from any continuous linear functional from which the functional can be recovered; i.e. a single vector can "store" all of the information contained in any entire continuous linear functional (mapping from ~H to CC).
|- (T e. (LinFn i^i ConFn) -> (`'bra` T) = U.{y e. ~H | A.x e. ~H (T` x) = (x .ih y)})
Theorem	cnvbracl 11682	Closure of the converse of the bra function.
|- (T e. (LinFn i^i ConFn) -> (`'bra` T) e. ~H)
Theorem	cnvbrabra 11683	The converse bra of the bra of a vector is the vector itself.
|- (A e. ~H -> (`'bra` (bra` A)) = A)
Theorem	bracnvbra 11684	The bra of the converse bra of a continuous linear functional.
|- (T e. (LinFn i^i ConFn) -> (bra` (`'bra` T)) = T)
Theorem	bracnlnval 11685	The vector that a continuous linear functional is the bra of.
|- (T e. (LinFn i^i ConFn) -> T = (bra` U.{y e. ~H | A.x e. ~H (T` x) = (x .ih y)}))
Theorem	cnvbramul 11686	Multiplication property of the converse bra function.
|- ((A e. CC /\ T e. (LinFn i^i ConFn)) -> (`'bra` (A .fn T)) = ((*` A) .h (`'bra` T)))
Theorem	kbass1 11687	Dirac bra-ket associative law ( | A>. <.B | ) | C>. = | A>.(<.B | C>.) i.e. the juxtaposition of an outer product with a ket equals a bra juxtaposed with an inner product. Since <.B | C>. is a complex number, it is the first argument in the inner product .h that it is mapped to, although in Dirac notation it is placed after the ket.
|- ((A e. ~H /\ B e. ~H /\ C e. ~H) -> ((A ketbra B)` C) = (((bra` B)` C) .h A))
Theorem	kbass2 11688	Dirac bra-ket associative law (<.A | B>.)<.C | = <.A | ( | B>. <.C | ) i.e. the juxtaposition of an inner product with a bra equals a ket juxtaposed with an outer product.
|- ((A e. ~H /\ B e. ~H /\ C e. ~H) -> (((bra` A)` B) .fn (bra` C)) = ((bra` A) o. (B ketbra C)))
Theorem	kbass3 11689	Dirac bra-ket associative law <.A | B>. <.C | D>. = (<.A | B>. <.C | ) | D>..
|- (((A e. ~H /\ B e. ~H) /\ (C e. ~H /\ D e. ~H)) -> (((bra` A)` B) x. ((bra` C)` D)) = ((((bra` A)` B) .fn (bra` C))` D))
Theorem	kbass4 11690	Dirac bra-ket associative law <.A | B>. <.C | D>. = <.A | ( | B>. <.C | D>.).
|- (((A e. ~H /\ B e. ~H) /\ (C e. ~H /\ D e. ~H)) -> (((bra` A)` B) x. ((bra` C)` D)) = ((bra` A)` (((bra` C)` D) .h B)))
Theorem	kbass5 11691	Dirac bra-ket associative law ( | A>. <.B | )( | C>. <.D | ) = (( | A>. <.B | ) | C>.)<.D |.
|- (((A e. ~H /\ B e. ~H) /\ (C e. ~H /\ D e. ~H)) -> ((A ketbra B) o. (C ketbra D)) = (((A ketbra B)` C) ketbra D))
Theorem	kbass6 11692	Dirac bra-ket associative law ( | A>. <.B | )( | C>. <.D | ) = | A>. (<.B | ( | C>. <.D | )).
|- (((A e. ~H /\ B e. ~H) /\ (C e. ~H /\ D e. ~H)) -> ((A ketbra B) o. (C ketbra D)) = (A ketbra (`'bra` ((bra` B) o. (C ketbra D)))))
Positive operators (cont.)
Theorem	leopg 11693	Ordering relation for positive operators. Definition of positive operator ordering in [Kreyszig] p. 470.
|- ((T e. A /\ U e. B) -> (T <_op U <-> ((U -op T) e. HrmOp /\ A.x e. ~H 0 <_ (((U -op T)` x) .ih x))))
Theorem	leop 11694	Ordering relation for operators. Definition of positive operator ordering in [Kreyszig] p. 470.
|- ((T e. HrmOp /\ U e. HrmOp) -> (T <_op U <-> A.x e. ~H 0 <_ (((U -op T)` x) .ih x)))
Theorem	leop2 11695	Ordering relation for operators. Definition of operator ordering in [Young] p. 141.
|- ((T e. HrmOp /\ U e. HrmOp) -> (T <_op U <-> A.x e. ~H ((T` x) .ih x) <_ ((U` x) .ih x)))
Theorem	leop3 11696	Operator ordering in terms of a positive operator. Definition of operator ordering in [Retherford] p. 49.
|- ((T e. HrmOp /\ U e. HrmOp) -> (T <_op U <-> 0hop <_op (U -op T)))
Theorem	leoppos 11697	Binary relation defining a positive operator. Definition VI.1 of [Retherford] p. 49.
|- (T e. HrmOp -> (0hop <_op T <-> A.x e. ~H 0 <_ ((T` x) .ih x)))
Theorem	leoprf2 11698	The ordering relation for operators is reflexive.
|- (T:~H-->~H -> T <_op T)
Theorem	leoprf 11699	The ordering relation for operators is reflexive.
|- (T e. HrmOp -> T <_op T)
Theorem	leopsq 11700	The square of a Hermitian operator is positive.
|- (T e. HrmOp -> 0hop <_op (T o. T))