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Theorem List for Metamath Proof Explorer - 31201-31300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremkbop 31201 The outer product of two vectors, expressed as ∣ 𝐴⟩⟨𝐡 ∣ in Dirac notation, is an operator. (Contributed by NM, 30-May-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ (𝐴 ketbra 𝐡): β„‹βŸΆ β„‹)
 
Theoremkbval 31202 The value of the operator resulting from the outer product ∣ 𝐴⟩ ⟨𝐡 ∣ of two vectors. Equation 8.1 of [Prugovecki] p. 376. (Contributed by NM, 15-May-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ ((𝐴 ketbra 𝐡)β€˜πΆ) = ((𝐢 Β·ih 𝐡) Β·β„Ž 𝐴))
 
Theoremkbmul 31203 Multiplication property of outer product. (Contributed by NM, 31-May-2006.) (New usage is discouraged.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ ((𝐴 Β·β„Ž 𝐡) ketbra 𝐢) = (𝐡 ketbra ((βˆ—β€˜π΄) Β·β„Ž 𝐢)))
 
Theoremkbpj 31204 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. (Contributed by NM, 30-May-2006.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ (normβ„Žβ€˜π΄) = 1) β†’ (𝐴 ketbra 𝐴) = (projβ„Žβ€˜(spanβ€˜{𝐴})))
 
Theoremeleigvec 31205* Membership in the set of eigenvectors of a Hilbert space operator. (Contributed by NM, 11-Mar-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
(𝑇: β„‹βŸΆ β„‹ β†’ (𝐴 ∈ (eigvecβ€˜π‘‡) ↔ (𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž ∧ βˆƒπ‘₯ ∈ β„‚ (π‘‡β€˜π΄) = (π‘₯ Β·β„Ž 𝐴))))
 
Theoremeleigvec2 31206 Membership in the set of eigenvectors of a Hilbert space operator. (Contributed by NM, 18-Mar-2006.) (New usage is discouraged.)
(𝑇: β„‹βŸΆ β„‹ β†’ (𝐴 ∈ (eigvecβ€˜π‘‡) ↔ (𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž ∧ (π‘‡β€˜π΄) ∈ (spanβ€˜{𝐴}))))
 
Theoremeleigveccl 31207 Closure of an eigenvector of a Hilbert space operator. (Contributed by NM, 23-Mar-2006.) (New usage is discouraged.)
((𝑇: β„‹βŸΆ β„‹ ∧ 𝐴 ∈ (eigvecβ€˜π‘‡)) β†’ 𝐴 ∈ β„‹)
 
Theoremeigvalval 31208 The eigenvalue of an eigenvector of a Hilbert space operator. (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.)
((𝑇: β„‹βŸΆ β„‹ ∧ 𝐴 ∈ (eigvecβ€˜π‘‡)) β†’ ((eigvalβ€˜π‘‡)β€˜π΄) = (((π‘‡β€˜π΄) Β·ih 𝐴) / ((normβ„Žβ€˜π΄)↑2)))
 
Theoremeigvalcl 31209 An eigenvalue is a complex number. (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.)
((𝑇: β„‹βŸΆ β„‹ ∧ 𝐴 ∈ (eigvecβ€˜π‘‡)) β†’ ((eigvalβ€˜π‘‡)β€˜π΄) ∈ β„‚)
 
Theoremeigvec1 31210 Property of an eigenvector. (Contributed by NM, 12-Mar-2006.) (New usage is discouraged.)
((𝑇: β„‹βŸΆ β„‹ ∧ 𝐴 ∈ (eigvecβ€˜π‘‡)) β†’ ((π‘‡β€˜π΄) = (((eigvalβ€˜π‘‡)β€˜π΄) Β·β„Ž 𝐴) ∧ 𝐴 β‰  0β„Ž))
 
Theoremeighmre 31211 The eigenvalues of a Hermitian operator are real. Equation 1.30 of [Hughes] p. 49. (Contributed by NM, 19-Mar-2006.) (New usage is discouraged.)
((𝑇 ∈ HrmOp ∧ 𝐴 ∈ (eigvecβ€˜π‘‡)) β†’ ((eigvalβ€˜π‘‡)β€˜π΄) ∈ ℝ)
 
Theoremeighmorth 31212 Eigenvectors of a Hermitian operator with distinct eigenvalues are orthogonal. Equation 1.31 of [Hughes] p. 49. (Contributed by NM, 23-Mar-2006.) (New usage is discouraged.)
(((𝑇 ∈ HrmOp ∧ 𝐴 ∈ (eigvecβ€˜π‘‡)) ∧ (𝐡 ∈ (eigvecβ€˜π‘‡) ∧ ((eigvalβ€˜π‘‡)β€˜π΄) β‰  ((eigvalβ€˜π‘‡)β€˜π΅))) β†’ (𝐴 Β·ih 𝐡) = 0)
 
Theoremnmopnegi 31213 Value of the norm of the negative of a Hilbert space operator. Unlike nmophmi 31279, the operator does not have to be bounded. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
𝑇: β„‹βŸΆ β„‹    β‡’   (normopβ€˜(-1 Β·op 𝑇)) = (normopβ€˜π‘‡)
 
Theoremlnop0 31214 The value of a linear Hilbert space operator at zero is zero. Remark in [Beran] p. 99. (Contributed by NM, 13-Aug-2006.) (New usage is discouraged.)
(𝑇 ∈ LinOp β†’ (π‘‡β€˜0β„Ž) = 0β„Ž)
 
Theoremlnopmul 31215 Multiplicative property of a linear Hilbert space operator. (Contributed by NM, 13-Aug-2006.) (New usage is discouraged.)
((𝑇 ∈ LinOp ∧ 𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‹) β†’ (π‘‡β€˜(𝐴 Β·β„Ž 𝐡)) = (𝐴 Β·β„Ž (π‘‡β€˜π΅)))
 
Theoremlnopli 31216 Basic scalar product property of a linear Hilbert space operator. (Contributed by NM, 23-Jan-2006.) (New usage is discouraged.)
𝑇 ∈ LinOp    β‡’   ((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ (π‘‡β€˜((𝐴 Β·β„Ž 𝐡) +β„Ž 𝐢)) = ((𝐴 Β·β„Ž (π‘‡β€˜π΅)) +β„Ž (π‘‡β€˜πΆ)))
 
Theoremlnopfi 31217 A linear Hilbert space operator is a Hilbert space operator. (Contributed by NM, 23-Jan-2006.) (New usage is discouraged.)
𝑇 ∈ LinOp    β‡’   π‘‡: β„‹βŸΆ β„‹
 
Theoremlnop0i 31218 The value of a linear Hilbert space operator at zero is zero. Remark in [Beran] p. 99. (Contributed by NM, 11-May-2005.) (New usage is discouraged.)
𝑇 ∈ LinOp    β‡’   (π‘‡β€˜0β„Ž) = 0β„Ž
 
Theoremlnopaddi 31219 Additive property of a linear Hilbert space operator. (Contributed by NM, 11-May-2005.) (New usage is discouraged.)
𝑇 ∈ LinOp    β‡’   ((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ (π‘‡β€˜(𝐴 +β„Ž 𝐡)) = ((π‘‡β€˜π΄) +β„Ž (π‘‡β€˜π΅)))
 
Theoremlnopmuli 31220 Multiplicative property of a linear Hilbert space operator. (Contributed by NM, 11-May-2005.) (New usage is discouraged.)
𝑇 ∈ LinOp    β‡’   ((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‹) β†’ (π‘‡β€˜(𝐴 Β·β„Ž 𝐡)) = (𝐴 Β·β„Ž (π‘‡β€˜π΅)))
 
Theoremlnopaddmuli 31221 Sum/product property of a linear Hilbert space operator. (Contributed by NM, 1-Jul-2005.) (New usage is discouraged.)
𝑇 ∈ LinOp    β‡’   ((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ (π‘‡β€˜(𝐡 +β„Ž (𝐴 Β·β„Ž 𝐢))) = ((π‘‡β€˜π΅) +β„Ž (𝐴 Β·β„Ž (π‘‡β€˜πΆ))))
 
Theoremlnopsubi 31222 Subtraction property for a linear Hilbert space operator. (Contributed by NM, 1-Jul-2005.) (New usage is discouraged.)
𝑇 ∈ LinOp    β‡’   ((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ (π‘‡β€˜(𝐴 βˆ’β„Ž 𝐡)) = ((π‘‡β€˜π΄) βˆ’β„Ž (π‘‡β€˜π΅)))
 
Theoremlnopsubmuli 31223 Subtraction/product property of a linear Hilbert space operator. (Contributed by NM, 2-Jul-2005.) (New usage is discouraged.)
𝑇 ∈ LinOp    β‡’   ((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ (π‘‡β€˜(𝐡 βˆ’β„Ž (𝐴 Β·β„Ž 𝐢))) = ((π‘‡β€˜π΅) βˆ’β„Ž (𝐴 Β·β„Ž (π‘‡β€˜πΆ))))
 
Theoremlnopmulsubi 31224 Product/subtraction property of a linear Hilbert space operator. (Contributed by NM, 2-Jul-2005.) (New usage is discouraged.)
𝑇 ∈ LinOp    β‡’   ((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ (π‘‡β€˜((𝐴 Β·β„Ž 𝐡) βˆ’β„Ž 𝐢)) = ((𝐴 Β·β„Ž (π‘‡β€˜π΅)) βˆ’β„Ž (π‘‡β€˜πΆ)))
 
Theoremhomco2 31225 Move a scalar product out of a composition of operators. The operator 𝑇 must be linear, unlike homco1 31049 that works for any operators. (Contributed by NM, 13-Aug-2006.) (New usage is discouraged.)
((𝐴 ∈ β„‚ ∧ 𝑇 ∈ LinOp ∧ π‘ˆ: β„‹βŸΆ β„‹) β†’ (𝑇 ∘ (𝐴 Β·op π‘ˆ)) = (𝐴 Β·op (𝑇 ∘ π‘ˆ)))
 
Theoremidunop 31226 The identity function (restricted to Hilbert space) is a unitary operator. (Contributed by NM, 21-Jan-2006.) (New usage is discouraged.)
( I β†Ύ β„‹) ∈ UniOp
 
Theorem0cnop 31227 The identically zero function is a continuous Hilbert space operator. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)
0hop ∈ ContOp
 
Theorem0cnfn 31228 The identically zero function is a continuous Hilbert space functional. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)
( β„‹ Γ— {0}) ∈ ContFn
 
Theoremidcnop 31229 The identity function (restricted to Hilbert space) is a continuous operator. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)
( I β†Ύ β„‹) ∈ ContOp
 
Theoremidhmop 31230 The Hilbert space identity operator is a Hermitian operator. (Contributed by NM, 22-Apr-2006.) (New usage is discouraged.)
Iop ∈ HrmOp
 
Theorem0hmop 31231 The identically zero function is a Hermitian operator. (Contributed by NM, 8-Aug-2006.) (New usage is discouraged.)
0hop ∈ HrmOp
 
Theorem0lnop 31232 The identically zero function is a linear Hilbert space operator. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)
0hop ∈ LinOp
 
Theorem0lnfn 31233 The identically zero function is a linear Hilbert space functional. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
( β„‹ Γ— {0}) ∈ LinFn
 
Theoremnmop0 31234 The norm of the zero operator is zero. (Contributed by NM, 8-Feb-2006.) (New usage is discouraged.)
(normopβ€˜ 0hop ) = 0
 
Theoremnmfn0 31235 The norm of the identically zero functional is zero. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.)
(normfnβ€˜( β„‹ Γ— {0})) = 0
 
TheoremhmopbdoptHIL 31236 A Hermitian operator is a bounded linear operator (Hellinger-Toeplitz Theorem). (Contributed by NM, 18-Jan-2008.) (New usage is discouraged.)
(𝑇 ∈ HrmOp β†’ 𝑇 ∈ BndLinOp)
 
Theoremhoddii 31237 Distributive law for Hilbert space operator difference. (Interestingly, the reverse distributive law hocsubdiri 31028 does not require linearity.) (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.)
𝑅 ∈ LinOp    &   π‘†: β„‹βŸΆ β„‹    &   π‘‡: β„‹βŸΆ β„‹    β‡’   (𝑅 ∘ (𝑆 βˆ’op 𝑇)) = ((𝑅 ∘ 𝑆) βˆ’op (𝑅 ∘ 𝑇))
 
Theoremhoddi 31238 Distributive law for Hilbert space operator difference. (Interestingly, the reverse distributive law hocsubdiri 31028 does not require linearity.) (Contributed by NM, 23-Aug-2006.) (New usage is discouraged.)
((𝑅 ∈ LinOp ∧ 𝑆: β„‹βŸΆ β„‹ ∧ 𝑇: β„‹βŸΆ β„‹) β†’ (𝑅 ∘ (𝑆 βˆ’op 𝑇)) = ((𝑅 ∘ 𝑆) βˆ’op (𝑅 ∘ 𝑇)))
 
Theoremnmop0h 31239 The norm of any operator on the trivial Hilbert space is zero. (This is the reason we need β„‹ β‰  0β„‹ in nmopun 31262.) (Contributed by NM, 24-Feb-2006.) (New usage is discouraged.)
(( β„‹ = 0β„‹ ∧ 𝑇: β„‹βŸΆ β„‹) β†’ (normopβ€˜π‘‡) = 0)
 
Theoremidlnop 31240 The identity function (restricted to Hilbert space) is a linear operator. (Contributed by NM, 24-Jan-2006.) (New usage is discouraged.)
( I β†Ύ β„‹) ∈ LinOp
 
Theorem0bdop 31241 The identically zero operator is bounded. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
0hop ∈ BndLinOp
 
Theoremadj0 31242 Adjoint of the zero operator. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.)
(adjβ„Žβ€˜ 0hop ) = 0hop
 
Theoremnmlnop0iALT 31243 A linear operator with a zero norm is identically zero. (Contributed by NM, 8-Feb-2006.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑇 ∈ LinOp    β‡’   ((normopβ€˜π‘‡) = 0 ↔ 𝑇 = 0hop )
 
Theoremnmlnop0iHIL 31244 A linear operator with a zero norm is identically zero. (Contributed by NM, 18-Jan-2008.) (New usage is discouraged.)
𝑇 ∈ LinOp    β‡’   ((normopβ€˜π‘‡) = 0 ↔ 𝑇 = 0hop )
 
Theoremnmlnopgt0i 31245 A linear Hilbert space operator that is not identically zero has a positive norm. (Contributed by NM, 9-Feb-2006.) (New usage is discouraged.)
𝑇 ∈ LinOp    β‡’   (𝑇 β‰  0hop ↔ 0 < (normopβ€˜π‘‡))
 
Theoremnmlnop0 31246 A linear operator with a zero norm is identically zero. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)
(𝑇 ∈ LinOp β†’ ((normopβ€˜π‘‡) = 0 ↔ 𝑇 = 0hop ))
 
Theoremnmlnopne0 31247 A linear operator with a nonzero norm is nonzero. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)
(𝑇 ∈ LinOp β†’ ((normopβ€˜π‘‡) β‰  0 ↔ 𝑇 β‰  0hop ))
 
Theoremlnopmi 31248 The scalar product of a linear operator is a linear operator. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
𝑇 ∈ LinOp    β‡’   (𝐴 ∈ β„‚ β†’ (𝐴 Β·op 𝑇) ∈ LinOp)
 
Theoremlnophsi 31249 The sum of two linear operators is linear. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
𝑆 ∈ LinOp    &   π‘‡ ∈ LinOp    β‡’   (𝑆 +op 𝑇) ∈ LinOp
 
Theoremlnophdi 31250 The difference of two linear operators is linear. (Contributed by NM, 27-Jul-2006.) (New usage is discouraged.)
𝑆 ∈ LinOp    &   π‘‡ ∈ LinOp    β‡’   (𝑆 βˆ’op 𝑇) ∈ LinOp
 
Theoremlnopcoi 31251 The composition of two linear operators is linear. (Contributed by NM, 8-Mar-2006.) (New usage is discouraged.)
𝑆 ∈ LinOp    &   π‘‡ ∈ LinOp    β‡’   (𝑆 ∘ 𝑇) ∈ LinOp
 
Theoremlnopco0i 31252 The composition of a linear operator with one whose norm is zero. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
𝑆 ∈ LinOp    &   π‘‡ ∈ LinOp    β‡’   ((normopβ€˜π‘‡) = 0 β†’ (normopβ€˜(𝑆 ∘ 𝑇)) = 0)
 
Theoremlnopeq0lem1 31253 Lemma for lnopeq0i 31255. Apply the generalized polarization identity polid2i 30405 to the quadratic form ((π‘‡β€˜π‘₯), π‘₯). (Contributed by NM, 26-Jul-2006.) (New usage is discouraged.)
𝑇 ∈ LinOp    &   π΄ ∈ β„‹    &   π΅ ∈ β„‹    β‡’   ((π‘‡β€˜π΄) Β·ih 𝐡) = (((((π‘‡β€˜(𝐴 +β„Ž 𝐡)) Β·ih (𝐴 +β„Ž 𝐡)) βˆ’ ((π‘‡β€˜(𝐴 βˆ’β„Ž 𝐡)) Β·ih (𝐴 βˆ’β„Ž 𝐡))) + (i Β· (((π‘‡β€˜(𝐴 +β„Ž (i Β·β„Ž 𝐡))) Β·ih (𝐴 +β„Ž (i Β·β„Ž 𝐡))) βˆ’ ((π‘‡β€˜(𝐴 βˆ’β„Ž (i Β·β„Ž 𝐡))) Β·ih (𝐴 βˆ’β„Ž (i Β·β„Ž 𝐡)))))) / 4)
 
Theoremlnopeq0lem2 31254 Lemma for lnopeq0i 31255. (Contributed by NM, 26-Jul-2006.) (New usage is discouraged.)
𝑇 ∈ LinOp    β‡’   ((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ ((π‘‡β€˜π΄) Β·ih 𝐡) = (((((π‘‡β€˜(𝐴 +β„Ž 𝐡)) Β·ih (𝐴 +β„Ž 𝐡)) βˆ’ ((π‘‡β€˜(𝐴 βˆ’β„Ž 𝐡)) Β·ih (𝐴 βˆ’β„Ž 𝐡))) + (i Β· (((π‘‡β€˜(𝐴 +β„Ž (i Β·β„Ž 𝐡))) Β·ih (𝐴 +β„Ž (i Β·β„Ž 𝐡))) βˆ’ ((π‘‡β€˜(𝐴 βˆ’β„Ž (i Β·β„Ž 𝐡))) Β·ih (𝐴 βˆ’β„Ž (i Β·β„Ž 𝐡)))))) / 4))
 
Theoremlnopeq0i 31255* A condition implying that a linear Hilbert space operator is identically zero. Unlike ho01i 31076 for arbitrary operators, when the operator is linear we need to consider only the values of the quadratic form (π‘‡β€˜π‘₯) Β·ih π‘₯). (Contributed by NM, 26-Jul-2006.) (New usage is discouraged.)
𝑇 ∈ LinOp    β‡’   (βˆ€π‘₯ ∈ β„‹ ((π‘‡β€˜π‘₯) Β·ih π‘₯) = 0 ↔ 𝑇 = 0hop )
 
Theoremlnopeqi 31256* Two linear Hilbert space operators are equal iff their quadratic forms are equal. (Contributed by NM, 27-Jul-2006.) (New usage is discouraged.)
𝑇 ∈ LinOp    &   π‘ˆ ∈ LinOp    β‡’   (βˆ€π‘₯ ∈ β„‹ ((π‘‡β€˜π‘₯) Β·ih π‘₯) = ((π‘ˆβ€˜π‘₯) Β·ih π‘₯) ↔ 𝑇 = π‘ˆ)
 
Theoremlnopeq 31257* Two linear Hilbert space operators are equal iff their quadratic forms are equal. (Contributed by NM, 27-Jul-2006.) (New usage is discouraged.)
((𝑇 ∈ LinOp ∧ π‘ˆ ∈ LinOp) β†’ (βˆ€π‘₯ ∈ β„‹ ((π‘‡β€˜π‘₯) Β·ih π‘₯) = ((π‘ˆβ€˜π‘₯) Β·ih π‘₯) ↔ 𝑇 = π‘ˆ))
 
Theoremlnopunilem1 31258* Lemma for lnopunii 31260. (Contributed by NM, 14-May-2005.) (New usage is discouraged.)
𝑇 ∈ LinOp    &   βˆ€π‘₯ ∈ β„‹ (normβ„Žβ€˜(π‘‡β€˜π‘₯)) = (normβ„Žβ€˜π‘₯)    &   π΄ ∈ β„‹    &   π΅ ∈ β„‹    &   πΆ ∈ β„‚    β‡’   (β„œβ€˜(𝐢 Β· ((π‘‡β€˜π΄) Β·ih (π‘‡β€˜π΅)))) = (β„œβ€˜(𝐢 Β· (𝐴 Β·ih 𝐡)))
 
Theoremlnopunilem2 31259* Lemma for lnopunii 31260. (Contributed by NM, 12-May-2005.) (New usage is discouraged.)
𝑇 ∈ LinOp    &   βˆ€π‘₯ ∈ β„‹ (normβ„Žβ€˜(π‘‡β€˜π‘₯)) = (normβ„Žβ€˜π‘₯)    &   π΄ ∈ β„‹    &   π΅ ∈ β„‹    β‡’   ((π‘‡β€˜π΄) Β·ih (π‘‡β€˜π΅)) = (𝐴 Β·ih 𝐡)
 
Theoremlnopunii 31260* If a linear operator (whose range is β„‹) 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.)
𝑇 ∈ LinOp    &   π‘‡: ℋ–ontoβ†’ β„‹    &   βˆ€π‘₯ ∈ β„‹ (normβ„Žβ€˜(π‘‡β€˜π‘₯)) = (normβ„Žβ€˜π‘₯)    β‡’   π‘‡ ∈ UniOp
 
Theoremelunop2 31261* 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.)
(𝑇 ∈ UniOp ↔ (𝑇 ∈ LinOp ∧ 𝑇: ℋ–ontoβ†’ β„‹ ∧ βˆ€π‘₯ ∈ β„‹ (normβ„Žβ€˜(π‘‡β€˜π‘₯)) = (normβ„Žβ€˜π‘₯)))
 
Theoremnmopun 31262 Norm of a unitary Hilbert space operator. (Contributed by NM, 25-Feb-2006.) (New usage is discouraged.)
(( β„‹ β‰  0β„‹ ∧ 𝑇 ∈ UniOp) β†’ (normopβ€˜π‘‡) = 1)
 
Theoremunopbd 31263 A unitary operator is a bounded linear operator. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
(𝑇 ∈ UniOp β†’ 𝑇 ∈ BndLinOp)
 
Theoremlnophmlem1 31264* Lemma for lnophmi 31266. (Contributed by NM, 24-Jan-2006.) (New usage is discouraged.)
𝐴 ∈ β„‹    &   π΅ ∈ β„‹    &   π‘‡ ∈ LinOp    &   βˆ€π‘₯ ∈ β„‹ (π‘₯ Β·ih (π‘‡β€˜π‘₯)) ∈ ℝ    β‡’   (𝐴 Β·ih (π‘‡β€˜π΄)) ∈ ℝ
 
Theoremlnophmlem2 31265* Lemma for lnophmi 31266. (Contributed by NM, 24-Jan-2006.) (New usage is discouraged.)
𝐴 ∈ β„‹    &   π΅ ∈ β„‹    &   π‘‡ ∈ LinOp    &   βˆ€π‘₯ ∈ β„‹ (π‘₯ Β·ih (π‘‡β€˜π‘₯)) ∈ ℝ    β‡’   (𝐴 Β·ih (π‘‡β€˜π΅)) = ((π‘‡β€˜π΄) Β·ih 𝐡)
 
Theoremlnophmi 31266* A linear operator is Hermitian if π‘₯ Β·ih (π‘‡β€˜π‘₯) takes only real values. Remark in [ReedSimon] p. 195. (Contributed by NM, 24-Jan-2006.) (New usage is discouraged.)
𝑇 ∈ LinOp    &   βˆ€π‘₯ ∈ β„‹ (π‘₯ Β·ih (π‘‡β€˜π‘₯)) ∈ ℝ    β‡’   π‘‡ ∈ HrmOp
 
Theoremlnophm 31267* A linear operator is Hermitian if π‘₯ Β·ih (π‘‡β€˜π‘₯) takes only real values. Remark in [ReedSimon] p. 195. (Contributed by NM, 24-Jan-2006.) (New usage is discouraged.)
((𝑇 ∈ LinOp ∧ βˆ€π‘₯ ∈ β„‹ (π‘₯ Β·ih (π‘‡β€˜π‘₯)) ∈ ℝ) β†’ 𝑇 ∈ HrmOp)
 
Theoremhmops 31268 The sum of two Hermitian operators is Hermitian. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
((𝑇 ∈ HrmOp ∧ π‘ˆ ∈ HrmOp) β†’ (𝑇 +op π‘ˆ) ∈ HrmOp)
 
Theoremhmopm 31269 The scalar product of a Hermitian operator with a real is Hermitian. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
((𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp) β†’ (𝐴 Β·op 𝑇) ∈ HrmOp)
 
Theoremhmopd 31270 The difference of two Hermitian operators is Hermitian. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
((𝑇 ∈ HrmOp ∧ π‘ˆ ∈ HrmOp) β†’ (𝑇 βˆ’op π‘ˆ) ∈ HrmOp)
 
Theoremhmopco 31271 The composition of two commuting Hermitian operators is Hermitian. (Contributed by NM, 22-Aug-2006.) (New usage is discouraged.)
((𝑇 ∈ HrmOp ∧ π‘ˆ ∈ HrmOp ∧ (𝑇 ∘ π‘ˆ) = (π‘ˆ ∘ 𝑇)) β†’ (𝑇 ∘ π‘ˆ) ∈ HrmOp)
 
Theoremnmbdoplbi 31272 A lower bound for the norm of a bounded linear operator. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
𝑇 ∈ BndLinOp    β‡’   (𝐴 ∈ β„‹ β†’ (normβ„Žβ€˜(π‘‡β€˜π΄)) ≀ ((normopβ€˜π‘‡) Β· (normβ„Žβ€˜π΄)))
 
Theoremnmbdoplb 31273 A lower bound for the norm of a bounded linear Hilbert space operator. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.)
((𝑇 ∈ BndLinOp ∧ 𝐴 ∈ β„‹) β†’ (normβ„Žβ€˜(π‘‡β€˜π΄)) ≀ ((normopβ€˜π‘‡) Β· (normβ„Žβ€˜π΄)))
 
Theoremnmcexi 31274* Lemma for nmcopexi 31275 and nmcfnexi 31299. 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.)
βˆƒπ‘¦ ∈ ℝ+ βˆ€π‘§ ∈ β„‹ ((normβ„Žβ€˜π‘§) < 𝑦 β†’ (π‘β€˜(π‘‡β€˜π‘§)) < 1)    &   (π‘†β€˜π‘‡) = sup({π‘š ∣ βˆƒπ‘₯ ∈ β„‹ ((normβ„Žβ€˜π‘₯) ≀ 1 ∧ π‘š = (π‘β€˜(π‘‡β€˜π‘₯)))}, ℝ*, < )    &   (π‘₯ ∈ β„‹ β†’ (π‘β€˜(π‘‡β€˜π‘₯)) ∈ ℝ)    &   (π‘β€˜(π‘‡β€˜0β„Ž)) = 0    &   (((𝑦 / 2) ∈ ℝ+ ∧ π‘₯ ∈ β„‹) β†’ ((𝑦 / 2) Β· (π‘β€˜(π‘‡β€˜π‘₯))) = (π‘β€˜(π‘‡β€˜((𝑦 / 2) Β·β„Ž π‘₯))))    β‡’   (π‘†β€˜π‘‡) ∈ ℝ
 
Theoremnmcopexi 31275 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.)
𝑇 ∈ LinOp    &   π‘‡ ∈ ContOp    β‡’   (normopβ€˜π‘‡) ∈ ℝ
 
Theoremnmcoplbi 31276 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.)
𝑇 ∈ LinOp    &   π‘‡ ∈ ContOp    β‡’   (𝐴 ∈ β„‹ β†’ (normβ„Žβ€˜(π‘‡β€˜π΄)) ≀ ((normopβ€˜π‘‡) Β· (normβ„Žβ€˜π΄)))
 
Theoremnmcopex 31277 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.)
((𝑇 ∈ LinOp ∧ 𝑇 ∈ ContOp) β†’ (normopβ€˜π‘‡) ∈ ℝ)
 
Theoremnmcoplb 31278 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.)
((𝑇 ∈ LinOp ∧ 𝑇 ∈ ContOp ∧ 𝐴 ∈ β„‹) β†’ (normβ„Žβ€˜(π‘‡β€˜π΄)) ≀ ((normopβ€˜π‘‡) Β· (normβ„Žβ€˜π΄)))
 
Theoremnmophmi 31279 The norm of the scalar product of a bounded linear operator. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
𝑇 ∈ BndLinOp    β‡’   (𝐴 ∈ β„‚ β†’ (normopβ€˜(𝐴 Β·op 𝑇)) = ((absβ€˜π΄) Β· (normopβ€˜π‘‡)))
 
Theorembdophmi 31280 The scalar product of a bounded linear operator is a bounded linear operator. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
𝑇 ∈ BndLinOp    β‡’   (𝐴 ∈ β„‚ β†’ (𝐴 Β·op 𝑇) ∈ BndLinOp)
 
Theoremlnconi 31281* Lemma for lnopconi 31282 and lnfnconi 31303. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)
(𝑇 ∈ 𝐢 β†’ 𝑆 ∈ ℝ)    &   ((𝑇 ∈ 𝐢 ∧ 𝑦 ∈ β„‹) β†’ (π‘β€˜(π‘‡β€˜π‘¦)) ≀ (𝑆 Β· (normβ„Žβ€˜π‘¦)))    &   (𝑇 ∈ 𝐢 ↔ βˆ€π‘₯ ∈ β„‹ βˆ€π‘§ ∈ ℝ+ βˆƒπ‘¦ ∈ ℝ+ βˆ€π‘€ ∈ β„‹ ((normβ„Žβ€˜(𝑀 βˆ’β„Ž π‘₯)) < 𝑦 β†’ (π‘β€˜((π‘‡β€˜π‘€)𝑀(π‘‡β€˜π‘₯))) < 𝑧))    &   (𝑦 ∈ β„‹ β†’ (π‘β€˜(π‘‡β€˜π‘¦)) ∈ ℝ)    &   ((𝑀 ∈ β„‹ ∧ π‘₯ ∈ β„‹) β†’ (π‘‡β€˜(𝑀 βˆ’β„Ž π‘₯)) = ((π‘‡β€˜π‘€)𝑀(π‘‡β€˜π‘₯)))    β‡’   (𝑇 ∈ 𝐢 ↔ βˆƒπ‘₯ ∈ ℝ βˆ€π‘¦ ∈ β„‹ (π‘β€˜(π‘‡β€˜π‘¦)) ≀ (π‘₯ Β· (normβ„Žβ€˜π‘¦)))
 
Theoremlnopconi 31282* A condition equivalent to "𝑇 is continuous" when 𝑇 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.)
𝑇 ∈ LinOp    β‡’   (𝑇 ∈ ContOp ↔ βˆƒπ‘₯ ∈ ℝ βˆ€π‘¦ ∈ β„‹ (normβ„Žβ€˜(π‘‡β€˜π‘¦)) ≀ (π‘₯ Β· (normβ„Žβ€˜π‘¦)))
 
Theoremlnopcon 31283* A condition equivalent to "𝑇 is continuous" when 𝑇 is linear. Theorem 3.5(iii) of [Beran] p. 99. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
(𝑇 ∈ LinOp β†’ (𝑇 ∈ ContOp ↔ βˆƒπ‘₯ ∈ ℝ βˆ€π‘¦ ∈ β„‹ (normβ„Žβ€˜(π‘‡β€˜π‘¦)) ≀ (π‘₯ Β· (normβ„Žβ€˜π‘¦))))
 
Theoremlnopcnbd 31284 A linear operator is continuous iff it is bounded. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
(𝑇 ∈ LinOp β†’ (𝑇 ∈ ContOp ↔ 𝑇 ∈ BndLinOp))
 
Theoremlncnopbd 31285 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.)
(𝑇 ∈ (LinOp ∩ ContOp) ↔ 𝑇 ∈ BndLinOp)
 
Theoremlncnbd 31286 A continuous linear operator is a bounded linear operator. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.)
(LinOp ∩ ContOp) = BndLinOp
 
Theoremlnopcnre 31287 A linear operator is continuous iff it is bounded. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
(𝑇 ∈ LinOp β†’ (𝑇 ∈ ContOp ↔ (normopβ€˜π‘‡) ∈ ℝ))
 
Theoremlnfnli 31288 Basic property of a linear Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)
𝑇 ∈ LinFn    β‡’   ((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ (π‘‡β€˜((𝐴 Β·β„Ž 𝐡) +β„Ž 𝐢)) = ((𝐴 Β· (π‘‡β€˜π΅)) + (π‘‡β€˜πΆ)))
 
Theoremlnfnfi 31289 A linear Hilbert space functional is a functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)
𝑇 ∈ LinFn    β‡’   π‘‡: β„‹βŸΆβ„‚
 
Theoremlnfn0i 31290 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.)
𝑇 ∈ LinFn    β‡’   (π‘‡β€˜0β„Ž) = 0
 
Theoremlnfnaddi 31291 Additive property of a linear Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)
𝑇 ∈ LinFn    β‡’   ((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ (π‘‡β€˜(𝐴 +β„Ž 𝐡)) = ((π‘‡β€˜π΄) + (π‘‡β€˜π΅)))
 
Theoremlnfnmuli 31292 Multiplicative property of a linear Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)
𝑇 ∈ LinFn    β‡’   ((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‹) β†’ (π‘‡β€˜(𝐴 Β·β„Ž 𝐡)) = (𝐴 Β· (π‘‡β€˜π΅)))
 
Theoremlnfnaddmuli 31293 Sum/product property of a linear Hilbert space functional. (Contributed by NM, 13-Feb-2006.) (New usage is discouraged.)
𝑇 ∈ LinFn    β‡’   ((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ (π‘‡β€˜(𝐡 +β„Ž (𝐴 Β·β„Ž 𝐢))) = ((π‘‡β€˜π΅) + (𝐴 Β· (π‘‡β€˜πΆ))))
 
Theoremlnfnsubi 31294 Subtraction property for a linear Hilbert space functional. (Contributed by NM, 13-Feb-2006.) (New usage is discouraged.)
𝑇 ∈ LinFn    β‡’   ((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ (π‘‡β€˜(𝐴 βˆ’β„Ž 𝐡)) = ((π‘‡β€˜π΄) βˆ’ (π‘‡β€˜π΅)))
 
Theoremlnfn0 31295 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.)
(𝑇 ∈ LinFn β†’ (π‘‡β€˜0β„Ž) = 0)
 
Theoremlnfnmul 31296 Multiplicative property of a linear Hilbert space functional. (Contributed by NM, 30-May-2006.) (New usage is discouraged.)
((𝑇 ∈ LinFn ∧ 𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‹) β†’ (π‘‡β€˜(𝐴 Β·β„Ž 𝐡)) = (𝐴 Β· (π‘‡β€˜π΅)))
 
Theoremnmbdfnlbi 31297 A lower bound for the norm of a bounded linear functional. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.)
(𝑇 ∈ LinFn ∧ (normfnβ€˜π‘‡) ∈ ℝ)    β‡’   (𝐴 ∈ β„‹ β†’ (absβ€˜(π‘‡β€˜π΄)) ≀ ((normfnβ€˜π‘‡) Β· (normβ„Žβ€˜π΄)))
 
Theoremnmbdfnlb 31298 A lower bound for the norm of a bounded linear functional. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.)
((𝑇 ∈ LinFn ∧ (normfnβ€˜π‘‡) ∈ ℝ ∧ 𝐴 ∈ β„‹) β†’ (absβ€˜(π‘‡β€˜π΄)) ≀ ((normfnβ€˜π‘‡) Β· (normβ„Žβ€˜π΄)))
 
Theoremnmcfnexi 31299 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.)
𝑇 ∈ LinFn    &   π‘‡ ∈ ContFn    β‡’   (normfnβ€˜π‘‡) ∈ ℝ
 
Theoremnmcfnlbi 31300 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.)
𝑇 ∈ LinFn    &   π‘‡ ∈ ContFn    β‡’   (𝐴 ∈ β„‹ β†’ (absβ€˜(π‘‡β€˜π΄)) ≀ ((normfnβ€˜π‘‡) Β· (normβ„Žβ€˜π΄)))
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206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 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