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Theorem List for Metamath Proof Explorer - 30701-30800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlnopaddmuli 30701 Sum/product property of a linear Hilbert space operator. (Contributed by NM, 1-Jul-2005.) (New usage is discouraged.)
๐‘‡ โˆˆ LinOp    โ‡’   ((๐ด โˆˆ โ„‚ โˆง ๐ต โˆˆ โ„‹ โˆง ๐ถ โˆˆ โ„‹) โ†’ (๐‘‡โ€˜(๐ต +โ„Ž (๐ด ยทโ„Ž ๐ถ))) = ((๐‘‡โ€˜๐ต) +โ„Ž (๐ด ยทโ„Ž (๐‘‡โ€˜๐ถ))))
 
Theoremlnopsubi 30702 Subtraction property for a linear Hilbert space operator. (Contributed by NM, 1-Jul-2005.) (New usage is discouraged.)
๐‘‡ โˆˆ LinOp    โ‡’   ((๐ด โˆˆ โ„‹ โˆง ๐ต โˆˆ โ„‹) โ†’ (๐‘‡โ€˜(๐ด โˆ’โ„Ž ๐ต)) = ((๐‘‡โ€˜๐ด) โˆ’โ„Ž (๐‘‡โ€˜๐ต)))
 
Theoremlnopsubmuli 30703 Subtraction/product property of a linear Hilbert space operator. (Contributed by NM, 2-Jul-2005.) (New usage is discouraged.)
๐‘‡ โˆˆ LinOp    โ‡’   ((๐ด โˆˆ โ„‚ โˆง ๐ต โˆˆ โ„‹ โˆง ๐ถ โˆˆ โ„‹) โ†’ (๐‘‡โ€˜(๐ต โˆ’โ„Ž (๐ด ยทโ„Ž ๐ถ))) = ((๐‘‡โ€˜๐ต) โˆ’โ„Ž (๐ด ยทโ„Ž (๐‘‡โ€˜๐ถ))))
 
Theoremlnopmulsubi 30704 Product/subtraction property of a linear Hilbert space operator. (Contributed by NM, 2-Jul-2005.) (New usage is discouraged.)
๐‘‡ โˆˆ LinOp    โ‡’   ((๐ด โˆˆ โ„‚ โˆง ๐ต โˆˆ โ„‹ โˆง ๐ถ โˆˆ โ„‹) โ†’ (๐‘‡โ€˜((๐ด ยทโ„Ž ๐ต) โˆ’โ„Ž ๐ถ)) = ((๐ด ยทโ„Ž (๐‘‡โ€˜๐ต)) โˆ’โ„Ž (๐‘‡โ€˜๐ถ)))
 
Theoremhomco2 30705 Move a scalar product out of a composition of operators. The operator ๐‘‡ must be linear, unlike homco1 30529 that works for any operators. (Contributed by NM, 13-Aug-2006.) (New usage is discouraged.)
((๐ด โˆˆ โ„‚ โˆง ๐‘‡ โˆˆ LinOp โˆง ๐‘ˆ: โ„‹โŸถ โ„‹) โ†’ (๐‘‡ โˆ˜ (๐ด ยทop ๐‘ˆ)) = (๐ด ยทop (๐‘‡ โˆ˜ ๐‘ˆ)))
 
Theoremidunop 30706 The identity function (restricted to Hilbert space) is a unitary operator. (Contributed by NM, 21-Jan-2006.) (New usage is discouraged.)
( I โ†พ โ„‹) โˆˆ UniOp
 
Theorem0cnop 30707 The identically zero function is a continuous Hilbert space operator. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)
0hop โˆˆ ContOp
 
Theorem0cnfn 30708 The identically zero function is a continuous Hilbert space functional. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)
( โ„‹ ร— {0}) โˆˆ ContFn
 
Theoremidcnop 30709 The identity function (restricted to Hilbert space) is a continuous operator. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)
( I โ†พ โ„‹) โˆˆ ContOp
 
Theoremidhmop 30710 The Hilbert space identity operator is a Hermitian operator. (Contributed by NM, 22-Apr-2006.) (New usage is discouraged.)
Iop โˆˆ HrmOp
 
Theorem0hmop 30711 The identically zero function is a Hermitian operator. (Contributed by NM, 8-Aug-2006.) (New usage is discouraged.)
0hop โˆˆ HrmOp
 
Theorem0lnop 30712 The identically zero function is a linear Hilbert space operator. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)
0hop โˆˆ LinOp
 
Theorem0lnfn 30713 The identically zero function is a linear Hilbert space functional. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
( โ„‹ ร— {0}) โˆˆ LinFn
 
Theoremnmop0 30714 The norm of the zero operator is zero. (Contributed by NM, 8-Feb-2006.) (New usage is discouraged.)
(normopโ€˜ 0hop ) = 0
 
Theoremnmfn0 30715 The norm of the identically zero functional is zero. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.)
(normfnโ€˜( โ„‹ ร— {0})) = 0
 
TheoremhmopbdoptHIL 30716 A Hermitian operator is a bounded linear operator (Hellinger-Toeplitz Theorem). (Contributed by NM, 18-Jan-2008.) (New usage is discouraged.)
(๐‘‡ โˆˆ HrmOp โ†’ ๐‘‡ โˆˆ BndLinOp)
 
Theoremhoddii 30717 Distributive law for Hilbert space operator difference. (Interestingly, the reverse distributive law hocsubdiri 30508 does not require linearity.) (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.)
๐‘… โˆˆ LinOp    &   ๐‘†: โ„‹โŸถ โ„‹    &   ๐‘‡: โ„‹โŸถ โ„‹    โ‡’   (๐‘… โˆ˜ (๐‘† โˆ’op ๐‘‡)) = ((๐‘… โˆ˜ ๐‘†) โˆ’op (๐‘… โˆ˜ ๐‘‡))
 
Theoremhoddi 30718 Distributive law for Hilbert space operator difference. (Interestingly, the reverse distributive law hocsubdiri 30508 does not require linearity.) (Contributed by NM, 23-Aug-2006.) (New usage is discouraged.)
((๐‘… โˆˆ LinOp โˆง ๐‘†: โ„‹โŸถ โ„‹ โˆง ๐‘‡: โ„‹โŸถ โ„‹) โ†’ (๐‘… โˆ˜ (๐‘† โˆ’op ๐‘‡)) = ((๐‘… โˆ˜ ๐‘†) โˆ’op (๐‘… โˆ˜ ๐‘‡)))
 
Theoremnmop0h 30719 The norm of any operator on the trivial Hilbert space is zero. (This is the reason we need โ„‹ โ‰  0โ„‹ in nmopun 30742.) (Contributed by NM, 24-Feb-2006.) (New usage is discouraged.)
(( โ„‹ = 0โ„‹ โˆง ๐‘‡: โ„‹โŸถ โ„‹) โ†’ (normopโ€˜๐‘‡) = 0)
 
Theoremidlnop 30720 The identity function (restricted to Hilbert space) is a linear operator. (Contributed by NM, 24-Jan-2006.) (New usage is discouraged.)
( I โ†พ โ„‹) โˆˆ LinOp
 
Theorem0bdop 30721 The identically zero operator is bounded. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
0hop โˆˆ BndLinOp
 
Theoremadj0 30722 Adjoint of the zero operator. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.)
(adjโ„Žโ€˜ 0hop ) = 0hop
 
Theoremnmlnop0iALT 30723 A linear operator with a zero norm is identically zero. (Contributed by NM, 8-Feb-2006.) (New usage is discouraged.) (Proof modification is discouraged.)
๐‘‡ โˆˆ LinOp    โ‡’   ((normopโ€˜๐‘‡) = 0 โ†” ๐‘‡ = 0hop )
 
Theoremnmlnop0iHIL 30724 A linear operator with a zero norm is identically zero. (Contributed by NM, 18-Jan-2008.) (New usage is discouraged.)
๐‘‡ โˆˆ LinOp    โ‡’   ((normopโ€˜๐‘‡) = 0 โ†” ๐‘‡ = 0hop )
 
Theoremnmlnopgt0i 30725 A linear Hilbert space operator that is not identically zero has a positive norm. (Contributed by NM, 9-Feb-2006.) (New usage is discouraged.)
๐‘‡ โˆˆ LinOp    โ‡’   (๐‘‡ โ‰  0hop โ†” 0 < (normopโ€˜๐‘‡))
 
Theoremnmlnop0 30726 A linear operator with a zero norm is identically zero. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)
(๐‘‡ โˆˆ LinOp โ†’ ((normopโ€˜๐‘‡) = 0 โ†” ๐‘‡ = 0hop ))
 
Theoremnmlnopne0 30727 A linear operator with a nonzero norm is nonzero. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)
(๐‘‡ โˆˆ LinOp โ†’ ((normopโ€˜๐‘‡) โ‰  0 โ†” ๐‘‡ โ‰  0hop ))
 
Theoremlnopmi 30728 The scalar product of a linear operator is a linear operator. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
๐‘‡ โˆˆ LinOp    โ‡’   (๐ด โˆˆ โ„‚ โ†’ (๐ด ยทop ๐‘‡) โˆˆ LinOp)
 
Theoremlnophsi 30729 The sum of two linear operators is linear. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
๐‘† โˆˆ LinOp    &   ๐‘‡ โˆˆ LinOp    โ‡’   (๐‘† +op ๐‘‡) โˆˆ LinOp
 
Theoremlnophdi 30730 The difference of two linear operators is linear. (Contributed by NM, 27-Jul-2006.) (New usage is discouraged.)
๐‘† โˆˆ LinOp    &   ๐‘‡ โˆˆ LinOp    โ‡’   (๐‘† โˆ’op ๐‘‡) โˆˆ LinOp
 
Theoremlnopcoi 30731 The composition of two linear operators is linear. (Contributed by NM, 8-Mar-2006.) (New usage is discouraged.)
๐‘† โˆˆ LinOp    &   ๐‘‡ โˆˆ LinOp    โ‡’   (๐‘† โˆ˜ ๐‘‡) โˆˆ LinOp
 
Theoremlnopco0i 30732 The composition of a linear operator with one whose norm is zero. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
๐‘† โˆˆ LinOp    &   ๐‘‡ โˆˆ LinOp    โ‡’   ((normopโ€˜๐‘‡) = 0 โ†’ (normopโ€˜(๐‘† โˆ˜ ๐‘‡)) = 0)
 
Theoremlnopeq0lem1 30733 Lemma for lnopeq0i 30735. Apply the generalized polarization identity polid2i 29885 to the quadratic form ((๐‘‡โ€˜๐‘ฅ), ๐‘ฅ). (Contributed by NM, 26-Jul-2006.) (New usage is discouraged.)
๐‘‡ โˆˆ LinOp    &   ๐ด โˆˆ โ„‹    &   ๐ต โˆˆ โ„‹    โ‡’   ((๐‘‡โ€˜๐ด) ยทih ๐ต) = (((((๐‘‡โ€˜(๐ด +โ„Ž ๐ต)) ยทih (๐ด +โ„Ž ๐ต)) โˆ’ ((๐‘‡โ€˜(๐ด โˆ’โ„Ž ๐ต)) ยทih (๐ด โˆ’โ„Ž ๐ต))) + (i ยท (((๐‘‡โ€˜(๐ด +โ„Ž (i ยทโ„Ž ๐ต))) ยทih (๐ด +โ„Ž (i ยทโ„Ž ๐ต))) โˆ’ ((๐‘‡โ€˜(๐ด โˆ’โ„Ž (i ยทโ„Ž ๐ต))) ยทih (๐ด โˆ’โ„Ž (i ยทโ„Ž ๐ต)))))) / 4)
 
Theoremlnopeq0lem2 30734 Lemma for lnopeq0i 30735. (Contributed by NM, 26-Jul-2006.) (New usage is discouraged.)
๐‘‡ โˆˆ LinOp    โ‡’   ((๐ด โˆˆ โ„‹ โˆง ๐ต โˆˆ โ„‹) โ†’ ((๐‘‡โ€˜๐ด) ยทih ๐ต) = (((((๐‘‡โ€˜(๐ด +โ„Ž ๐ต)) ยทih (๐ด +โ„Ž ๐ต)) โˆ’ ((๐‘‡โ€˜(๐ด โˆ’โ„Ž ๐ต)) ยทih (๐ด โˆ’โ„Ž ๐ต))) + (i ยท (((๐‘‡โ€˜(๐ด +โ„Ž (i ยทโ„Ž ๐ต))) ยทih (๐ด +โ„Ž (i ยทโ„Ž ๐ต))) โˆ’ ((๐‘‡โ€˜(๐ด โˆ’โ„Ž (i ยทโ„Ž ๐ต))) ยทih (๐ด โˆ’โ„Ž (i ยทโ„Ž ๐ต)))))) / 4))
 
Theoremlnopeq0i 30735* A condition implying that a linear Hilbert space operator is identically zero. Unlike ho01i 30556 for arbitrary operators, when the operator is linear we need to consider only the values of the quadratic form (๐‘‡โ€˜๐‘ฅ) ยทih ๐‘ฅ). (Contributed by NM, 26-Jul-2006.) (New usage is discouraged.)
๐‘‡ โˆˆ LinOp    โ‡’   (โˆ€๐‘ฅ โˆˆ โ„‹ ((๐‘‡โ€˜๐‘ฅ) ยทih ๐‘ฅ) = 0 โ†” ๐‘‡ = 0hop )
 
Theoremlnopeqi 30736* Two linear Hilbert space operators are equal iff their quadratic forms are equal. (Contributed by NM, 27-Jul-2006.) (New usage is discouraged.)
๐‘‡ โˆˆ LinOp    &   ๐‘ˆ โˆˆ LinOp    โ‡’   (โˆ€๐‘ฅ โˆˆ โ„‹ ((๐‘‡โ€˜๐‘ฅ) ยทih ๐‘ฅ) = ((๐‘ˆโ€˜๐‘ฅ) ยทih ๐‘ฅ) โ†” ๐‘‡ = ๐‘ˆ)
 
Theoremlnopeq 30737* Two linear Hilbert space operators are equal iff their quadratic forms are equal. (Contributed by NM, 27-Jul-2006.) (New usage is discouraged.)
((๐‘‡ โˆˆ LinOp โˆง ๐‘ˆ โˆˆ LinOp) โ†’ (โˆ€๐‘ฅ โˆˆ โ„‹ ((๐‘‡โ€˜๐‘ฅ) ยทih ๐‘ฅ) = ((๐‘ˆโ€˜๐‘ฅ) ยทih ๐‘ฅ) โ†” ๐‘‡ = ๐‘ˆ))
 
Theoremlnopunilem1 30738* Lemma for lnopunii 30740. (Contributed by NM, 14-May-2005.) (New usage is discouraged.)
๐‘‡ โˆˆ LinOp    &   โˆ€๐‘ฅ โˆˆ โ„‹ (normโ„Žโ€˜(๐‘‡โ€˜๐‘ฅ)) = (normโ„Žโ€˜๐‘ฅ)    &   ๐ด โˆˆ โ„‹    &   ๐ต โˆˆ โ„‹    &   ๐ถ โˆˆ โ„‚    โ‡’   (โ„œโ€˜(๐ถ ยท ((๐‘‡โ€˜๐ด) ยทih (๐‘‡โ€˜๐ต)))) = (โ„œโ€˜(๐ถ ยท (๐ด ยทih ๐ต)))
 
Theoremlnopunilem2 30739* Lemma for lnopunii 30740. (Contributed by NM, 12-May-2005.) (New usage is discouraged.)
๐‘‡ โˆˆ LinOp    &   โˆ€๐‘ฅ โˆˆ โ„‹ (normโ„Žโ€˜(๐‘‡โ€˜๐‘ฅ)) = (normโ„Žโ€˜๐‘ฅ)    &   ๐ด โˆˆ โ„‹    &   ๐ต โˆˆ โ„‹    โ‡’   ((๐‘‡โ€˜๐ด) ยทih (๐‘‡โ€˜๐ต)) = (๐ด ยทih ๐ต)
 
Theoremlnopunii 30740* If a linear operator (whose range is โ„‹) is idempotent in the norm, the operator is unitary. Similar to theorem in [AkhiezerGlazman] p. 73. (Contributed by NM, 23-Jan-2006.) (New usage is discouraged.)
๐‘‡ โˆˆ LinOp    &   ๐‘‡: โ„‹โ€“ontoโ†’ โ„‹    &   โˆ€๐‘ฅ โˆˆ โ„‹ (normโ„Žโ€˜(๐‘‡โ€˜๐‘ฅ)) = (normโ„Žโ€˜๐‘ฅ)    โ‡’   ๐‘‡ โˆˆ UniOp
 
Theoremelunop2 30741* An operator is unitary iff it is linear, onto, and idempotent in the norm. Similar to theorem in [AkhiezerGlazman] p. 73, and its converse. (Contributed by NM, 24-Feb-2006.) (New usage is discouraged.)
(๐‘‡ โˆˆ UniOp โ†” (๐‘‡ โˆˆ LinOp โˆง ๐‘‡: โ„‹โ€“ontoโ†’ โ„‹ โˆง โˆ€๐‘ฅ โˆˆ โ„‹ (normโ„Žโ€˜(๐‘‡โ€˜๐‘ฅ)) = (normโ„Žโ€˜๐‘ฅ)))
 
Theoremnmopun 30742 Norm of a unitary Hilbert space operator. (Contributed by NM, 25-Feb-2006.) (New usage is discouraged.)
(( โ„‹ โ‰  0โ„‹ โˆง ๐‘‡ โˆˆ UniOp) โ†’ (normopโ€˜๐‘‡) = 1)
 
Theoremunopbd 30743 A unitary operator is a bounded linear operator. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
(๐‘‡ โˆˆ UniOp โ†’ ๐‘‡ โˆˆ BndLinOp)
 
Theoremlnophmlem1 30744* Lemma for lnophmi 30746. (Contributed by NM, 24-Jan-2006.) (New usage is discouraged.)
๐ด โˆˆ โ„‹    &   ๐ต โˆˆ โ„‹    &   ๐‘‡ โˆˆ LinOp    &   โˆ€๐‘ฅ โˆˆ โ„‹ (๐‘ฅ ยทih (๐‘‡โ€˜๐‘ฅ)) โˆˆ โ„    โ‡’   (๐ด ยทih (๐‘‡โ€˜๐ด)) โˆˆ โ„
 
Theoremlnophmlem2 30745* Lemma for lnophmi 30746. (Contributed by NM, 24-Jan-2006.) (New usage is discouraged.)
๐ด โˆˆ โ„‹    &   ๐ต โˆˆ โ„‹    &   ๐‘‡ โˆˆ LinOp    &   โˆ€๐‘ฅ โˆˆ โ„‹ (๐‘ฅ ยทih (๐‘‡โ€˜๐‘ฅ)) โˆˆ โ„    โ‡’   (๐ด ยทih (๐‘‡โ€˜๐ต)) = ((๐‘‡โ€˜๐ด) ยทih ๐ต)
 
Theoremlnophmi 30746* A linear operator is Hermitian if ๐‘ฅ ยทih (๐‘‡โ€˜๐‘ฅ) takes only real values. Remark in [ReedSimon] p. 195. (Contributed by NM, 24-Jan-2006.) (New usage is discouraged.)
๐‘‡ โˆˆ LinOp    &   โˆ€๐‘ฅ โˆˆ โ„‹ (๐‘ฅ ยทih (๐‘‡โ€˜๐‘ฅ)) โˆˆ โ„    โ‡’   ๐‘‡ โˆˆ HrmOp
 
Theoremlnophm 30747* A linear operator is Hermitian if ๐‘ฅ ยทih (๐‘‡โ€˜๐‘ฅ) takes only real values. Remark in [ReedSimon] p. 195. (Contributed by NM, 24-Jan-2006.) (New usage is discouraged.)
((๐‘‡ โˆˆ LinOp โˆง โˆ€๐‘ฅ โˆˆ โ„‹ (๐‘ฅ ยทih (๐‘‡โ€˜๐‘ฅ)) โˆˆ โ„) โ†’ ๐‘‡ โˆˆ HrmOp)
 
Theoremhmops 30748 The sum of two Hermitian operators is Hermitian. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
((๐‘‡ โˆˆ HrmOp โˆง ๐‘ˆ โˆˆ HrmOp) โ†’ (๐‘‡ +op ๐‘ˆ) โˆˆ HrmOp)
 
Theoremhmopm 30749 The scalar product of a Hermitian operator with a real is Hermitian. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
((๐ด โˆˆ โ„ โˆง ๐‘‡ โˆˆ HrmOp) โ†’ (๐ด ยทop ๐‘‡) โˆˆ HrmOp)
 
Theoremhmopd 30750 The difference of two Hermitian operators is Hermitian. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
((๐‘‡ โˆˆ HrmOp โˆง ๐‘ˆ โˆˆ HrmOp) โ†’ (๐‘‡ โˆ’op ๐‘ˆ) โˆˆ HrmOp)
 
Theoremhmopco 30751 The composition of two commuting Hermitian operators is Hermitian. (Contributed by NM, 22-Aug-2006.) (New usage is discouraged.)
((๐‘‡ โˆˆ HrmOp โˆง ๐‘ˆ โˆˆ HrmOp โˆง (๐‘‡ โˆ˜ ๐‘ˆ) = (๐‘ˆ โˆ˜ ๐‘‡)) โ†’ (๐‘‡ โˆ˜ ๐‘ˆ) โˆˆ HrmOp)
 
Theoremnmbdoplbi 30752 A lower bound for the norm of a bounded linear operator. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
๐‘‡ โˆˆ BndLinOp    โ‡’   (๐ด โˆˆ โ„‹ โ†’ (normโ„Žโ€˜(๐‘‡โ€˜๐ด)) โ‰ค ((normopโ€˜๐‘‡) ยท (normโ„Žโ€˜๐ด)))
 
Theoremnmbdoplb 30753 A lower bound for the norm of a bounded linear Hilbert space operator. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.)
((๐‘‡ โˆˆ BndLinOp โˆง ๐ด โˆˆ โ„‹) โ†’ (normโ„Žโ€˜(๐‘‡โ€˜๐ด)) โ‰ค ((normopโ€˜๐‘‡) ยท (normโ„Žโ€˜๐ด)))
 
Theoremnmcexi 30754* Lemma for nmcopexi 30755 and nmcfnexi 30779. The norm of a continuous linear Hilbert space operator or functional exists. Theorem 3.5(i) of [Beran] p. 99. (Contributed by Mario Carneiro, 17-Nov-2013.) (Proof shortened by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
โˆƒ๐‘ฆ โˆˆ โ„+ โˆ€๐‘ง โˆˆ โ„‹ ((normโ„Žโ€˜๐‘ง) < ๐‘ฆ โ†’ (๐‘โ€˜(๐‘‡โ€˜๐‘ง)) < 1)    &   (๐‘†โ€˜๐‘‡) = sup({๐‘š โˆฃ โˆƒ๐‘ฅ โˆˆ โ„‹ ((normโ„Žโ€˜๐‘ฅ) โ‰ค 1 โˆง ๐‘š = (๐‘โ€˜(๐‘‡โ€˜๐‘ฅ)))}, โ„*, < )    &   (๐‘ฅ โˆˆ โ„‹ โ†’ (๐‘โ€˜(๐‘‡โ€˜๐‘ฅ)) โˆˆ โ„)    &   (๐‘โ€˜(๐‘‡โ€˜0โ„Ž)) = 0    &   (((๐‘ฆ / 2) โˆˆ โ„+ โˆง ๐‘ฅ โˆˆ โ„‹) โ†’ ((๐‘ฆ / 2) ยท (๐‘โ€˜(๐‘‡โ€˜๐‘ฅ))) = (๐‘โ€˜(๐‘‡โ€˜((๐‘ฆ / 2) ยทโ„Ž ๐‘ฅ))))    โ‡’   (๐‘†โ€˜๐‘‡) โˆˆ โ„
 
Theoremnmcopexi 30755 The norm of a continuous linear Hilbert space operator exists. Theorem 3.5(i) of [Beran] p. 99. (Contributed by NM, 5-Feb-2006.) (Proof shortened by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.)
๐‘‡ โˆˆ LinOp    &   ๐‘‡ โˆˆ ContOp    โ‡’   (normopโ€˜๐‘‡) โˆˆ โ„
 
Theoremnmcoplbi 30756 A lower bound for the norm of a continuous linear operator. Theorem 3.5(ii) of [Beran] p. 99. (Contributed by NM, 7-Feb-2006.) (Revised by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.)
๐‘‡ โˆˆ LinOp    &   ๐‘‡ โˆˆ ContOp    โ‡’   (๐ด โˆˆ โ„‹ โ†’ (normโ„Žโ€˜(๐‘‡โ€˜๐ด)) โ‰ค ((normopโ€˜๐‘‡) ยท (normโ„Žโ€˜๐ด)))
 
Theoremnmcopex 30757 The norm of a continuous linear Hilbert space operator exists. Theorem 3.5(i) of [Beran] p. 99. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)
((๐‘‡ โˆˆ LinOp โˆง ๐‘‡ โˆˆ ContOp) โ†’ (normopโ€˜๐‘‡) โˆˆ โ„)
 
Theoremnmcoplb 30758 A lower bound for the norm of a continuous linear Hilbert space operator. Theorem 3.5(ii) of [Beran] p. 99. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)
((๐‘‡ โˆˆ LinOp โˆง ๐‘‡ โˆˆ ContOp โˆง ๐ด โˆˆ โ„‹) โ†’ (normโ„Žโ€˜(๐‘‡โ€˜๐ด)) โ‰ค ((normopโ€˜๐‘‡) ยท (normโ„Žโ€˜๐ด)))
 
Theoremnmophmi 30759 The norm of the scalar product of a bounded linear operator. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
๐‘‡ โˆˆ BndLinOp    โ‡’   (๐ด โˆˆ โ„‚ โ†’ (normopโ€˜(๐ด ยทop ๐‘‡)) = ((absโ€˜๐ด) ยท (normopโ€˜๐‘‡)))
 
Theorembdophmi 30760 The scalar product of a bounded linear operator is a bounded linear operator. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
๐‘‡ โˆˆ BndLinOp    โ‡’   (๐ด โˆˆ โ„‚ โ†’ (๐ด ยทop ๐‘‡) โˆˆ BndLinOp)
 
Theoremlnconi 30761* Lemma for lnopconi 30762 and lnfnconi 30783. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)
(๐‘‡ โˆˆ ๐ถ โ†’ ๐‘† โˆˆ โ„)    &   ((๐‘‡ โˆˆ ๐ถ โˆง ๐‘ฆ โˆˆ โ„‹) โ†’ (๐‘โ€˜(๐‘‡โ€˜๐‘ฆ)) โ‰ค (๐‘† ยท (normโ„Žโ€˜๐‘ฆ)))    &   (๐‘‡ โˆˆ ๐ถ โ†” โˆ€๐‘ฅ โˆˆ โ„‹ โˆ€๐‘ง โˆˆ โ„+ โˆƒ๐‘ฆ โˆˆ โ„+ โˆ€๐‘ค โˆˆ โ„‹ ((normโ„Žโ€˜(๐‘ค โˆ’โ„Ž ๐‘ฅ)) < ๐‘ฆ โ†’ (๐‘โ€˜((๐‘‡โ€˜๐‘ค)๐‘€(๐‘‡โ€˜๐‘ฅ))) < ๐‘ง))    &   (๐‘ฆ โˆˆ โ„‹ โ†’ (๐‘โ€˜(๐‘‡โ€˜๐‘ฆ)) โˆˆ โ„)    &   ((๐‘ค โˆˆ โ„‹ โˆง ๐‘ฅ โˆˆ โ„‹) โ†’ (๐‘‡โ€˜(๐‘ค โˆ’โ„Ž ๐‘ฅ)) = ((๐‘‡โ€˜๐‘ค)๐‘€(๐‘‡โ€˜๐‘ฅ)))    โ‡’   (๐‘‡ โˆˆ ๐ถ โ†” โˆƒ๐‘ฅ โˆˆ โ„ โˆ€๐‘ฆ โˆˆ โ„‹ (๐‘โ€˜(๐‘‡โ€˜๐‘ฆ)) โ‰ค (๐‘ฅ ยท (normโ„Žโ€˜๐‘ฆ)))
 
Theoremlnopconi 30762* A condition equivalent to "๐‘‡ is continuous" when ๐‘‡ is linear. Theorem 3.5(iii) of [Beran] p. 99. (Contributed by NM, 7-Feb-2006.) (Proof shortened by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.)
๐‘‡ โˆˆ LinOp    โ‡’   (๐‘‡ โˆˆ ContOp โ†” โˆƒ๐‘ฅ โˆˆ โ„ โˆ€๐‘ฆ โˆˆ โ„‹ (normโ„Žโ€˜(๐‘‡โ€˜๐‘ฆ)) โ‰ค (๐‘ฅ ยท (normโ„Žโ€˜๐‘ฆ)))
 
Theoremlnopcon 30763* A condition equivalent to "๐‘‡ is continuous" when ๐‘‡ is linear. Theorem 3.5(iii) of [Beran] p. 99. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
(๐‘‡ โˆˆ LinOp โ†’ (๐‘‡ โˆˆ ContOp โ†” โˆƒ๐‘ฅ โˆˆ โ„ โˆ€๐‘ฆ โˆˆ โ„‹ (normโ„Žโ€˜(๐‘‡โ€˜๐‘ฆ)) โ‰ค (๐‘ฅ ยท (normโ„Žโ€˜๐‘ฆ))))
 
Theoremlnopcnbd 30764 A linear operator is continuous iff it is bounded. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
(๐‘‡ โˆˆ LinOp โ†’ (๐‘‡ โˆˆ ContOp โ†” ๐‘‡ โˆˆ BndLinOp))
 
Theoremlncnopbd 30765 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. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.)
(๐‘‡ โˆˆ (LinOp โˆฉ ContOp) โ†” ๐‘‡ โˆˆ BndLinOp)
 
Theoremlncnbd 30766 A continuous linear operator is a bounded linear operator. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.)
(LinOp โˆฉ ContOp) = BndLinOp
 
Theoremlnopcnre 30767 A linear operator is continuous iff it is bounded. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
(๐‘‡ โˆˆ LinOp โ†’ (๐‘‡ โˆˆ ContOp โ†” (normopโ€˜๐‘‡) โˆˆ โ„))
 
Theoremlnfnli 30768 Basic property of a linear Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)
๐‘‡ โˆˆ LinFn    โ‡’   ((๐ด โˆˆ โ„‚ โˆง ๐ต โˆˆ โ„‹ โˆง ๐ถ โˆˆ โ„‹) โ†’ (๐‘‡โ€˜((๐ด ยทโ„Ž ๐ต) +โ„Ž ๐ถ)) = ((๐ด ยท (๐‘‡โ€˜๐ต)) + (๐‘‡โ€˜๐ถ)))
 
Theoremlnfnfi 30769 A linear Hilbert space functional is a functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)
๐‘‡ โˆˆ LinFn    โ‡’   ๐‘‡: โ„‹โŸถโ„‚
 
Theoremlnfn0i 30770 The value of a linear Hilbert space functional at zero is zero. Remark in [Beran] p. 99. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)
๐‘‡ โˆˆ LinFn    โ‡’   (๐‘‡โ€˜0โ„Ž) = 0
 
Theoremlnfnaddi 30771 Additive property of a linear Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)
๐‘‡ โˆˆ LinFn    โ‡’   ((๐ด โˆˆ โ„‹ โˆง ๐ต โˆˆ โ„‹) โ†’ (๐‘‡โ€˜(๐ด +โ„Ž ๐ต)) = ((๐‘‡โ€˜๐ด) + (๐‘‡โ€˜๐ต)))
 
Theoremlnfnmuli 30772 Multiplicative property of a linear Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)
๐‘‡ โˆˆ LinFn    โ‡’   ((๐ด โˆˆ โ„‚ โˆง ๐ต โˆˆ โ„‹) โ†’ (๐‘‡โ€˜(๐ด ยทโ„Ž ๐ต)) = (๐ด ยท (๐‘‡โ€˜๐ต)))
 
Theoremlnfnaddmuli 30773 Sum/product property of a linear Hilbert space functional. (Contributed by NM, 13-Feb-2006.) (New usage is discouraged.)
๐‘‡ โˆˆ LinFn    โ‡’   ((๐ด โˆˆ โ„‚ โˆง ๐ต โˆˆ โ„‹ โˆง ๐ถ โˆˆ โ„‹) โ†’ (๐‘‡โ€˜(๐ต +โ„Ž (๐ด ยทโ„Ž ๐ถ))) = ((๐‘‡โ€˜๐ต) + (๐ด ยท (๐‘‡โ€˜๐ถ))))
 
Theoremlnfnsubi 30774 Subtraction property for a linear Hilbert space functional. (Contributed by NM, 13-Feb-2006.) (New usage is discouraged.)
๐‘‡ โˆˆ LinFn    โ‡’   ((๐ด โˆˆ โ„‹ โˆง ๐ต โˆˆ โ„‹) โ†’ (๐‘‡โ€˜(๐ด โˆ’โ„Ž ๐ต)) = ((๐‘‡โ€˜๐ด) โˆ’ (๐‘‡โ€˜๐ต)))
 
Theoremlnfn0 30775 The value of a linear Hilbert space functional at zero is zero. Remark in [Beran] p. 99. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.)
(๐‘‡ โˆˆ LinFn โ†’ (๐‘‡โ€˜0โ„Ž) = 0)
 
Theoremlnfnmul 30776 Multiplicative property of a linear Hilbert space functional. (Contributed by NM, 30-May-2006.) (New usage is discouraged.)
((๐‘‡ โˆˆ LinFn โˆง ๐ด โˆˆ โ„‚ โˆง ๐ต โˆˆ โ„‹) โ†’ (๐‘‡โ€˜(๐ด ยทโ„Ž ๐ต)) = (๐ด ยท (๐‘‡โ€˜๐ต)))
 
Theoremnmbdfnlbi 30777 A lower bound for the norm of a bounded linear functional. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.)
(๐‘‡ โˆˆ LinFn โˆง (normfnโ€˜๐‘‡) โˆˆ โ„)    โ‡’   (๐ด โˆˆ โ„‹ โ†’ (absโ€˜(๐‘‡โ€˜๐ด)) โ‰ค ((normfnโ€˜๐‘‡) ยท (normโ„Žโ€˜๐ด)))
 
Theoremnmbdfnlb 30778 A lower bound for the norm of a bounded linear functional. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.)
((๐‘‡ โˆˆ LinFn โˆง (normfnโ€˜๐‘‡) โˆˆ โ„ โˆง ๐ด โˆˆ โ„‹) โ†’ (absโ€˜(๐‘‡โ€˜๐ด)) โ‰ค ((normfnโ€˜๐‘‡) ยท (normโ„Žโ€˜๐ด)))
 
Theoremnmcfnexi 30779 The norm of a continuous linear Hilbert space functional exists. Theorem 3.5(i) of [Beran] p. 99. (Contributed by NM, 14-Feb-2006.) (Proof shortened by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.)
๐‘‡ โˆˆ LinFn    &   ๐‘‡ โˆˆ ContFn    โ‡’   (normfnโ€˜๐‘‡) โˆˆ โ„
 
Theoremnmcfnlbi 30780 A lower bound for the norm of a continuous linear functional. Theorem 3.5(ii) of [Beran] p. 99. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
๐‘‡ โˆˆ LinFn    &   ๐‘‡ โˆˆ ContFn    โ‡’   (๐ด โˆˆ โ„‹ โ†’ (absโ€˜(๐‘‡โ€˜๐ด)) โ‰ค ((normfnโ€˜๐‘‡) ยท (normโ„Žโ€˜๐ด)))
 
Theoremnmcfnex 30781 The norm of a continuous linear Hilbert space functional exists. Theorem 3.5(i) of [Beran] p. 99. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
((๐‘‡ โˆˆ LinFn โˆง ๐‘‡ โˆˆ ContFn) โ†’ (normfnโ€˜๐‘‡) โˆˆ โ„)
 
Theoremnmcfnlb 30782 A lower bound of the norm of a continuous linear Hilbert space functional. Theorem 3.5(ii) of [Beran] p. 99. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
((๐‘‡ โˆˆ LinFn โˆง ๐‘‡ โˆˆ ContFn โˆง ๐ด โˆˆ โ„‹) โ†’ (absโ€˜(๐‘‡โ€˜๐ด)) โ‰ค ((normfnโ€˜๐‘‡) ยท (normโ„Žโ€˜๐ด)))
 
Theoremlnfnconi 30783* A condition equivalent to "๐‘‡ is continuous" when ๐‘‡ is linear. Theorem 3.5(iii) of [Beran] p. 99. (Contributed by NM, 14-Feb-2006.) (Proof shortened by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.)
๐‘‡ โˆˆ LinFn    โ‡’   (๐‘‡ โˆˆ ContFn โ†” โˆƒ๐‘ฅ โˆˆ โ„ โˆ€๐‘ฆ โˆˆ โ„‹ (absโ€˜(๐‘‡โ€˜๐‘ฆ)) โ‰ค (๐‘ฅ ยท (normโ„Žโ€˜๐‘ฆ)))
 
Theoremlnfncon 30784* A condition equivalent to "๐‘‡ is continuous" when ๐‘‡ is linear. Theorem 3.5(iii) of [Beran] p. 99. (Contributed by NM, 16-Feb-2006.) (New usage is discouraged.)
(๐‘‡ โˆˆ LinFn โ†’ (๐‘‡ โˆˆ ContFn โ†” โˆƒ๐‘ฅ โˆˆ โ„ โˆ€๐‘ฆ โˆˆ โ„‹ (absโ€˜(๐‘‡โ€˜๐‘ฆ)) โ‰ค (๐‘ฅ ยท (normโ„Žโ€˜๐‘ฆ))))
 
Theoremlnfncnbd 30785 A linear functional is continuous iff it is bounded. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.)
(๐‘‡ โˆˆ LinFn โ†’ (๐‘‡ โˆˆ ContFn โ†” (normfnโ€˜๐‘‡) โˆˆ โ„))
 
Theoremimaelshi 30786 The image of a subspace under a linear operator is a subspace. (Contributed by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
๐‘‡ โˆˆ LinOp    &   ๐ด โˆˆ Sโ„‹    โ‡’   (๐‘‡ โ€œ ๐ด) โˆˆ Sโ„‹
 
Theoremrnelshi 30787 The range of a linear operator is a subspace. (Contributed by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.)
๐‘‡ โˆˆ LinOp    โ‡’   ran ๐‘‡ โˆˆ Sโ„‹
 
Theoremnlelshi 30788 The null space of a linear functional is a subspace. (Contributed by NM, 11-Feb-2006.) (Revised by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.)
๐‘‡ โˆˆ LinFn    โ‡’   (nullโ€˜๐‘‡) โˆˆ Sโ„‹
 
Theoremnlelchi 30789 The null space of a continuous linear functional is a closed subspace. Remark 3.8 of [Beran] p. 103. (Contributed by NM, 11-Feb-2006.) (Proof shortened by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
๐‘‡ โˆˆ LinFn    &   ๐‘‡ โˆˆ ContFn    โ‡’   (nullโ€˜๐‘‡) โˆˆ Cโ„‹
 
20.6.11  Riesz lemma
 
Theoremriesz3i 30790* A continuous linear functional can be expressed as an inner product. Existence part of Theorem 3.9 of [Beran] p. 104. (Contributed by NM, 13-Feb-2006.) (New usage is discouraged.)
๐‘‡ โˆˆ LinFn    &   ๐‘‡ โˆˆ ContFn    โ‡’   โˆƒ๐‘ค โˆˆ โ„‹ โˆ€๐‘ฃ โˆˆ โ„‹ (๐‘‡โ€˜๐‘ฃ) = (๐‘ฃ ยทih ๐‘ค)
 
Theoremriesz4i 30791* A continuous linear functional can be expressed as an inner product. Uniqueness part of Theorem 3.9 of [Beran] p. 104. (Contributed by NM, 13-Feb-2006.) (New usage is discouraged.)
๐‘‡ โˆˆ LinFn    &   ๐‘‡ โˆˆ ContFn    โ‡’   โˆƒ!๐‘ค โˆˆ โ„‹ โˆ€๐‘ฃ โˆˆ โ„‹ (๐‘‡โ€˜๐‘ฃ) = (๐‘ฃ ยทih ๐‘ค)
 
Theoremriesz4 30792* A continuous linear functional can be expressed as an inner product. Uniqueness part of Theorem 3.9 of [Beran] p. 104. See riesz2 30794 for the bounded linear functional version. (Contributed by NM, 16-Feb-2006.) (New usage is discouraged.)
(๐‘‡ โˆˆ (LinFn โˆฉ ContFn) โ†’ โˆƒ!๐‘ค โˆˆ โ„‹ โˆ€๐‘ฃ โˆˆ โ„‹ (๐‘‡โ€˜๐‘ฃ) = (๐‘ฃ ยทih ๐‘ค))
 
Theoremriesz1 30793* 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 30794. For the continuous linear functional version, see riesz3i 30790 and riesz4 30792. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.)
(๐‘‡ โˆˆ LinFn โ†’ ((normfnโ€˜๐‘‡) โˆˆ โ„ โ†” โˆƒ๐‘ฆ โˆˆ โ„‹ โˆ€๐‘ฅ โˆˆ โ„‹ (๐‘‡โ€˜๐‘ฅ) = (๐‘ฅ ยทih ๐‘ฆ)))
 
Theoremriesz2 30794* 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 30793. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.)
((๐‘‡ โˆˆ LinFn โˆง (normfnโ€˜๐‘‡) โˆˆ โ„) โ†’ โˆƒ!๐‘ฆ โˆˆ โ„‹ โˆ€๐‘ฅ โˆˆ โ„‹ (๐‘‡โ€˜๐‘ฅ) = (๐‘ฅ ยทih ๐‘ฆ))
 
20.6.12  Adjoints (cont.)
 
Theoremcnlnadjlem1 30795* Lemma for cnlnadji 30804 (Theorem 3.10 of [Beran] p. 104: every continuous linear operator has an adjoint). The value of the auxiliary functional ๐บ. (Contributed by NM, 16-Feb-2006.) (New usage is discouraged.)
๐‘‡ โˆˆ LinOp    &   ๐‘‡ โˆˆ ContOp    &   ๐บ = (๐‘” โˆˆ โ„‹ โ†ฆ ((๐‘‡โ€˜๐‘”) ยทih ๐‘ฆ))    โ‡’   (๐ด โˆˆ โ„‹ โ†’ (๐บโ€˜๐ด) = ((๐‘‡โ€˜๐ด) ยทih ๐‘ฆ))
 
Theoremcnlnadjlem2 30796* Lemma for cnlnadji 30804. ๐บ is a continuous linear functional. (Contributed by NM, 16-Feb-2006.) (New usage is discouraged.)
๐‘‡ โˆˆ LinOp    &   ๐‘‡ โˆˆ ContOp    &   ๐บ = (๐‘” โˆˆ โ„‹ โ†ฆ ((๐‘‡โ€˜๐‘”) ยทih ๐‘ฆ))    โ‡’   (๐‘ฆ โˆˆ โ„‹ โ†’ (๐บ โˆˆ LinFn โˆง ๐บ โˆˆ ContFn))
 
Theoremcnlnadjlem3 30797* Lemma for cnlnadji 30804. By riesz4 30792, ๐ต is the unique vector such that (๐‘‡โ€˜๐‘ฃ) ยทih ๐‘ฆ) = (๐‘ฃ ยทih ๐‘ค) for all ๐‘ฃ. (Contributed by NM, 17-Feb-2006.) (New usage is discouraged.)
๐‘‡ โˆˆ LinOp    &   ๐‘‡ โˆˆ ContOp    &   ๐บ = (๐‘” โˆˆ โ„‹ โ†ฆ ((๐‘‡โ€˜๐‘”) ยทih ๐‘ฆ))    &   ๐ต = (โ„ฉ๐‘ค โˆˆ โ„‹ โˆ€๐‘ฃ โˆˆ โ„‹ ((๐‘‡โ€˜๐‘ฃ) ยทih ๐‘ฆ) = (๐‘ฃ ยทih ๐‘ค))    โ‡’   (๐‘ฆ โˆˆ โ„‹ โ†’ ๐ต โˆˆ โ„‹)
 
Theoremcnlnadjlem4 30798* Lemma for cnlnadji 30804. The values of auxiliary function ๐น are vectors. (Contributed by NM, 17-Feb-2006.) (Proof shortened by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.)
๐‘‡ โˆˆ LinOp    &   ๐‘‡ โˆˆ ContOp    &   ๐บ = (๐‘” โˆˆ โ„‹ โ†ฆ ((๐‘‡โ€˜๐‘”) ยทih ๐‘ฆ))    &   ๐ต = (โ„ฉ๐‘ค โˆˆ โ„‹ โˆ€๐‘ฃ โˆˆ โ„‹ ((๐‘‡โ€˜๐‘ฃ) ยทih ๐‘ฆ) = (๐‘ฃ ยทih ๐‘ค))    &   ๐น = (๐‘ฆ โˆˆ โ„‹ โ†ฆ ๐ต)    โ‡’   (๐ด โˆˆ โ„‹ โ†’ (๐นโ€˜๐ด) โˆˆ โ„‹)
 
Theoremcnlnadjlem5 30799* Lemma for cnlnadji 30804. ๐น is an adjoint of ๐‘‡ (later, we will show it is unique). (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.)
๐‘‡ โˆˆ LinOp    &   ๐‘‡ โˆˆ ContOp    &   ๐บ = (๐‘” โˆˆ โ„‹ โ†ฆ ((๐‘‡โ€˜๐‘”) ยทih ๐‘ฆ))    &   ๐ต = (โ„ฉ๐‘ค โˆˆ โ„‹ โˆ€๐‘ฃ โˆˆ โ„‹ ((๐‘‡โ€˜๐‘ฃ) ยทih ๐‘ฆ) = (๐‘ฃ ยทih ๐‘ค))    &   ๐น = (๐‘ฆ โˆˆ โ„‹ โ†ฆ ๐ต)    โ‡’   ((๐ด โˆˆ โ„‹ โˆง ๐ถ โˆˆ โ„‹) โ†’ ((๐‘‡โ€˜๐ถ) ยทih ๐ด) = (๐ถ ยทih (๐นโ€˜๐ด)))
 
Theoremcnlnadjlem6 30800* Lemma for cnlnadji 30804. ๐น is linear. (Contributed by NM, 17-Feb-2006.) (New usage is discouraged.)
๐‘‡ โˆˆ LinOp    &   ๐‘‡ โˆˆ ContOp    &   ๐บ = (๐‘” โˆˆ โ„‹ โ†ฆ ((๐‘‡โ€˜๐‘”) ยทih ๐‘ฆ))    &   ๐ต = (โ„ฉ๐‘ค โˆˆ โ„‹ โˆ€๐‘ฃ โˆˆ โ„‹ ((๐‘‡โ€˜๐‘ฃ) ยทih ๐‘ฆ) = (๐‘ฃ ยทih ๐‘ค))    &   ๐น = (๐‘ฆ โˆˆ โ„‹ โ†ฆ ๐ต)    โ‡’   ๐น โˆˆ LinOp
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