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Theorem List for Metamath Proof Explorer - 28901-29000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlnopl 28901 Basic linearity property of a linear Hilbert space operator. (Contributed by NM, 22-Jan-2006.) (New usage is discouraged.)
(((𝑇 ∈ LinOp ∧ 𝐴 ∈ ℂ) ∧ (𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ)) → (𝑇‘((𝐴 · 𝐵) + 𝐶)) = ((𝐴 · (𝑇𝐵)) + (𝑇𝐶)))

Theoremunop 28902 Basic inner product property of a unitary operator. (Contributed by NM, 22-Jan-2006.) (New usage is discouraged.)
((𝑇 ∈ UniOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝑇𝐴) ·ih (𝑇𝐵)) = (𝐴 ·ih 𝐵))

Theoremunopf1o 28903 A unitary operator in Hilbert space is one-to-one and onto. (Contributed by NM, 22-Jan-2006.) (New usage is discouraged.)
(𝑇 ∈ UniOp → 𝑇: ℋ–1-1-onto→ ℋ)

Theoremunopnorm 28904 A unitary operator is idempotent in the norm. (Contributed by NM, 25-Feb-2006.) (New usage is discouraged.)
((𝑇 ∈ UniOp ∧ 𝐴 ∈ ℋ) → (norm‘(𝑇𝐴)) = (norm𝐴))

Theoremcnvunop 28905 The inverse (converse) of a unitary operator in Hilbert space is unitary. Theorem in [AkhiezerGlazman] p. 72. (Contributed by NM, 22-Jan-2006.) (New usage is discouraged.)
(𝑇 ∈ UniOp → 𝑇 ∈ UniOp)

Theoremunopadj 28906 The inverse (converse) of a unitary operator is its adjoint. Equation 2 of [AkhiezerGlazman] p. 72. (Contributed by NM, 22-Jan-2006.) (New usage is discouraged.)
((𝑇 ∈ UniOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝑇𝐴) ·ih 𝐵) = (𝐴 ·ih (𝑇𝐵)))

Theoremunoplin 28907 A unitary operator is linear. Theorem in [AkhiezerGlazman] p. 72. (Contributed by NM, 22-Jan-2006.) (New usage is discouraged.)
(𝑇 ∈ UniOp → 𝑇 ∈ LinOp)

Theoremcounop 28908 The composition of two unitary operators is unitary. (Contributed by NM, 22-Jan-2006.) (New usage is discouraged.)
((𝑆 ∈ UniOp ∧ 𝑇 ∈ UniOp) → (𝑆𝑇) ∈ UniOp)

Theoremhmop 28909 Basic inner product property of a Hermitian operator. (Contributed by NM, 19-Mar-2006.) (New usage is discouraged.)
((𝑇 ∈ HrmOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih (𝑇𝐵)) = ((𝑇𝐴) ·ih 𝐵))

Theoremhmopre 28910 The inner product of the value and argument of a Hermitian operator is real. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
((𝑇 ∈ HrmOp ∧ 𝐴 ∈ ℋ) → ((𝑇𝐴) ·ih 𝐴) ∈ ℝ)

Theoremnmfnlb 28911 A lower bound for a functional norm. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ ∧ (norm𝐴) ≤ 1) → (abs‘(𝑇𝐴)) ≤ (normfn𝑇))

Theoremnmfnleub 28912* An upper bound for the norm of a functional. (Contributed by NM, 24-May-2006.) (Revised by Mario Carneiro, 7-Sep-2014.) (New usage is discouraged.)
((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℝ*) → ((normfn𝑇) ≤ 𝐴 ↔ ∀𝑥 ∈ ℋ ((norm𝑥) ≤ 1 → (abs‘(𝑇𝑥)) ≤ 𝐴)))

Theoremnmfnleub2 28913* An upper bound for the norm of a functional. (Contributed by NM, 24-May-2006.) (New usage is discouraged.)
((𝑇: ℋ⟶ℂ ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ ∀𝑥 ∈ ℋ (abs‘(𝑇𝑥)) ≤ (𝐴 · (norm𝑥))) → (normfn𝑇) ≤ 𝐴)

Theoremnmfnge0 28914 The norm of any Hilbert space functional is nonnegative. (Contributed by NM, 24-May-2006.) (New usage is discouraged.)
(𝑇: ℋ⟶ℂ → 0 ≤ (normfn𝑇))

Theoremelnlfn 28915 Membership in the null space of a Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (Revised by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.)
(𝑇: ℋ⟶ℂ → (𝐴 ∈ (null‘𝑇) ↔ (𝐴 ∈ ℋ ∧ (𝑇𝐴) = 0)))

Theoremelnlfn2 28916 Membership in the null space of a Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (Revised by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.)
((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ (null‘𝑇)) → (𝑇𝐴) = 0)

Theoremcnfnc 28917* Basic continuity property of a continuous functional. (Contributed by NM, 11-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
((𝑇 ∈ ContFn ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℝ+) → ∃𝑥 ∈ ℝ+𝑦 ∈ ℋ ((norm‘(𝑦 𝐴)) < 𝑥 → (abs‘((𝑇𝑦) − (𝑇𝐴))) < 𝐵))

Theoremlnfnl 28918 Basic linearity property of a linear functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)
(((𝑇 ∈ LinFn ∧ 𝐴 ∈ ℂ) ∧ (𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ)) → (𝑇‘((𝐴 · 𝐵) + 𝐶)) = ((𝐴 · (𝑇𝐵)) + (𝑇𝐶)))

Theoremadjcl 28919 Closure of the adjoint of a Hilbert space operator. (Contributed by NM, 17-Jun-2006.) (New usage is discouraged.)

Theoremadj1 28920 Property of an adjoint Hilbert space operator. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)
((𝑇 ∈ dom adj𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih (𝑇𝐵)) = (((adj𝑇)‘𝐴) ·ih 𝐵))

Theoremadj2 28921 Property of an adjoint Hilbert space operator. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)
((𝑇 ∈ dom adj𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝑇𝐴) ·ih 𝐵) = (𝐴 ·ih ((adj𝑇)‘𝐵)))

Theoremadjeq 28922* A property that determines the adjoint of a Hilbert space operator. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.)
((𝑇: ℋ⟶ ℋ ∧ 𝑆: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑆𝑦))) → (adj𝑇) = 𝑆)

Theoremadjadj 28923 Double adjoint. Theorem 3.11(iv) of [Beran] p. 106. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)

Theoremadjvalval 28924* Value of the value of the adjoint function. (Contributed by NM, 22-Feb-2006.) (Proof shortened by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.)
((𝑇 ∈ dom adj𝐴 ∈ ℋ) → ((adj𝑇)‘𝐴) = (𝑤 ∈ ℋ ∀𝑥 ∈ ℋ ((𝑇𝑥) ·ih 𝐴) = (𝑥 ·ih 𝑤)))

Theoremunopadj2 28925 The adjoint of a unitary operator is its inverse (converse). Equation 2 of [AkhiezerGlazman] p. 72. (Contributed by NM, 23-Feb-2006.) (New usage is discouraged.)
(𝑇 ∈ UniOp → (adj𝑇) = 𝑇)

Theoremhmopadj 28926 A Hermitian operator is self-adjoint. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)
(𝑇 ∈ HrmOp → (adj𝑇) = 𝑇)

Theoremhmdmadj 28927 Every Hermitian operator has an adjoint. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)
(𝑇 ∈ HrmOp → 𝑇 ∈ dom adj)

Theoremhmopadj2 28928 An operator is Hermitian iff it is self-adjoint. Definition of Hermitian in [Halmos] p. 41. (Contributed by NM, 9-Apr-2006.) (New usage is discouraged.)
(𝑇 ∈ dom adj → (𝑇 ∈ HrmOp ↔ (adj𝑇) = 𝑇))

Theoremhmoplin 28929 A Hermitian operator is linear. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)
(𝑇 ∈ HrmOp → 𝑇 ∈ LinOp)

Theorembrafval 28930* The bra of a vector, expressed as 𝐴 in Dirac notation. See df-bra 28837. (Contributed by NM, 15-May-2006.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)
(𝐴 ∈ ℋ → (bra‘𝐴) = (𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝐴)))

Theorembraval 28931 A bra-ket juxtaposition, expressed as 𝐴𝐵 in Dirac notation, equals the inner product of the vectors. Based on definition of bra in [Prugovecki] p. 186. (Contributed by NM, 15-May-2006.) (Revised by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((bra‘𝐴)‘𝐵) = (𝐵 ·ih 𝐴))

Theorembraadd 28932 Linearity property of bra for addition. (Contributed by NM, 23-May-2006.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((bra‘𝐴)‘(𝐵 + 𝐶)) = (((bra‘𝐴)‘𝐵) + ((bra‘𝐴)‘𝐶)))

Theorembramul 28933 Linearity property of bra for multiplication. (Contributed by NM, 23-May-2006.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((bra‘𝐴)‘(𝐵 · 𝐶)) = (𝐵 · ((bra‘𝐴)‘𝐶)))

Theorembrafn 28934 The bra function is a functional. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
(𝐴 ∈ ℋ → (bra‘𝐴): ℋ⟶ℂ)

Theorembralnfn 28935 The Dirac bra function is a linear functional. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
(𝐴 ∈ ℋ → (bra‘𝐴) ∈ LinFn)

Theorembracl 28936 Closure of the bra function. (Contributed by NM, 23-May-2006.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((bra‘𝐴)‘𝐵) ∈ ℂ)

Theorembra0 28937 The Dirac bra of the zero vector. (Contributed by NM, 25-May-2006.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)
(bra‘0) = ( ℋ × {0})

Theorembrafnmul 28938 Anti-linearity property of bra functional for multiplication. (Contributed by NM, 31-May-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (bra‘(𝐴 · 𝐵)) = ((∗‘𝐴) ·fn (bra‘𝐵)))

Theoremkbfval 28939* The outer product of two vectors, expressed as 𝐴 𝐵 in Dirac notation. See df-kb 28838. (Contributed by NM, 15-May-2006.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ketbra 𝐵) = (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝐵) · 𝐴)))

Theoremkbop 28940 The outer product of two vectors, expressed as 𝐴 𝐵 in Dirac notation, is an operator. (Contributed by NM, 30-May-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ketbra 𝐵): ℋ⟶ ℋ)

Theoremkbval 28941 The value of the operator resulting from the outer product 𝐴 𝐵 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 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ketbra 𝐵)‘𝐶) = ((𝐶 ·ih 𝐵) · 𝐴))

Theoremkbmul 28942 Multiplication property of outer product. (Contributed by NM, 31-May-2006.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 · 𝐵) ketbra 𝐶) = (𝐵 ketbra ((∗‘𝐴) · 𝐶)))

Theoremkbpj 28943 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 ∈ ℋ ∧ (norm𝐴) = 1) → (𝐴 ketbra 𝐴) = (proj‘(span‘{𝐴})))

Theoremeleigvec 28944* 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: ℋ⟶ ℋ → (𝐴 ∈ (eigvec‘𝑇) ↔ (𝐴 ∈ ℋ ∧ 𝐴 ≠ 0 ∧ ∃𝑥 ∈ ℂ (𝑇𝐴) = (𝑥 · 𝐴))))

Theoremeleigvec2 28945 Membership in the set of eigenvectors of a Hilbert space operator. (Contributed by NM, 18-Mar-2006.) (New usage is discouraged.)
(𝑇: ℋ⟶ ℋ → (𝐴 ∈ (eigvec‘𝑇) ↔ (𝐴 ∈ ℋ ∧ 𝐴 ≠ 0 ∧ (𝑇𝐴) ∈ (span‘{𝐴}))))

Theoremeleigveccl 28946 Closure of an eigenvector of a Hilbert space operator. (Contributed by NM, 23-Mar-2006.) (New usage is discouraged.)
((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ (eigvec‘𝑇)) → 𝐴 ∈ ℋ)

Theoremeigvalval 28947 The eigenvalue of an eigenvector of a Hilbert space operator. (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.)
((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ (eigvec‘𝑇)) → ((eigval‘𝑇)‘𝐴) = (((𝑇𝐴) ·ih 𝐴) / ((norm𝐴)↑2)))

Theoremeigvalcl 28948 An eigenvalue is a complex number. (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.)
((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ (eigvec‘𝑇)) → ((eigval‘𝑇)‘𝐴) ∈ ℂ)

Theoremeigvec1 28949 Property of an eigenvector. (Contributed by NM, 12-Mar-2006.) (New usage is discouraged.)
((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ (eigvec‘𝑇)) → ((𝑇𝐴) = (((eigval‘𝑇)‘𝐴) · 𝐴) ∧ 𝐴 ≠ 0))

Theoremeighmre 28950 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 ∈ (eigvec‘𝑇)) → ((eigval‘𝑇)‘𝐴) ∈ ℝ)

Theoremeighmorth 28951 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 ∈ (eigvec‘𝑇)) ∧ (𝐵 ∈ (eigvec‘𝑇) ∧ ((eigval‘𝑇)‘𝐴) ≠ ((eigval‘𝑇)‘𝐵))) → (𝐴 ·ih 𝐵) = 0)

Theoremnmopnegi 28952 Value of the norm of the negative of a Hilbert space operator. Unlike nmophmi 29018, the operator does not have to be bounded. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
𝑇: ℋ⟶ ℋ       (normop‘(-1 ·op 𝑇)) = (normop𝑇)

Theoremlnop0 28953 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 → (𝑇‘0) = 0)

Theoremlnopmul 28954 Multiplicative property of a linear Hilbert space operator. (Contributed by NM, 13-Aug-2006.) (New usage is discouraged.)
((𝑇 ∈ LinOp ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (𝑇‘(𝐴 · 𝐵)) = (𝐴 · (𝑇𝐵)))

Theoremlnopli 28955 Basic scalar product property of a linear Hilbert space operator. (Contributed by NM, 23-Jan-2006.) (New usage is discouraged.)
𝑇 ∈ LinOp       ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝑇‘((𝐴 · 𝐵) + 𝐶)) = ((𝐴 · (𝑇𝐵)) + (𝑇𝐶)))

Theoremlnopfi 28956 A linear Hilbert space operator is a Hilbert space operator. (Contributed by NM, 23-Jan-2006.) (New usage is discouraged.)
𝑇 ∈ LinOp       𝑇: ℋ⟶ ℋ

Theoremlnop0i 28957 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       (𝑇‘0) = 0

Theoremlnopaddi 28958 Additive property of a linear Hilbert space operator. (Contributed by NM, 11-May-2005.) (New usage is discouraged.)
𝑇 ∈ LinOp       ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘(𝐴 + 𝐵)) = ((𝑇𝐴) + (𝑇𝐵)))

Theoremlnopmuli 28959 Multiplicative property of a linear Hilbert space operator. (Contributed by NM, 11-May-2005.) (New usage is discouraged.)
𝑇 ∈ LinOp       ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (𝑇‘(𝐴 · 𝐵)) = (𝐴 · (𝑇𝐵)))

Theoremlnopaddmuli 28960 Sum/product property of a linear Hilbert space operator. (Contributed by NM, 1-Jul-2005.) (New usage is discouraged.)
𝑇 ∈ LinOp       ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝑇‘(𝐵 + (𝐴 · 𝐶))) = ((𝑇𝐵) + (𝐴 · (𝑇𝐶))))

Theoremlnopsubi 28961 Subtraction property for a linear Hilbert space operator. (Contributed by NM, 1-Jul-2005.) (New usage is discouraged.)
𝑇 ∈ LinOp       ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘(𝐴 𝐵)) = ((𝑇𝐴) − (𝑇𝐵)))

Theoremlnopsubmuli 28962 Subtraction/product property of a linear Hilbert space operator. (Contributed by NM, 2-Jul-2005.) (New usage is discouraged.)
𝑇 ∈ LinOp       ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝑇‘(𝐵 (𝐴 · 𝐶))) = ((𝑇𝐵) − (𝐴 · (𝑇𝐶))))

Theoremlnopmulsubi 28963 Product/subtraction property of a linear Hilbert space operator. (Contributed by NM, 2-Jul-2005.) (New usage is discouraged.)
𝑇 ∈ LinOp       ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝑇‘((𝐴 · 𝐵) − 𝐶)) = ((𝐴 · (𝑇𝐵)) − (𝑇𝐶)))

Theoremhomco2 28964 Move a scalar product out of a composition of operators. The operator 𝑇 must be linear, unlike homco1 28788 that works for any operators. (Contributed by NM, 13-Aug-2006.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝑇 ∈ LinOp ∧ 𝑈: ℋ⟶ ℋ) → (𝑇 ∘ (𝐴 ·op 𝑈)) = (𝐴 ·op (𝑇𝑈)))

Theoremidunop 28965 The identity function (restricted to Hilbert space) is a unitary operator. (Contributed by NM, 21-Jan-2006.) (New usage is discouraged.)
( I ↾ ℋ) ∈ UniOp

Theorem0cnop 28966 The identically zero function is a continuous Hilbert space operator. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)
0hop ∈ ContOp

Theorem0cnfn 28967 The identically zero function is a continuous Hilbert space functional. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)
( ℋ × {0}) ∈ ContFn

Theoremidcnop 28968 The identity function (restricted to Hilbert space) is a continuous operator. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)
( I ↾ ℋ) ∈ ContOp

Theoremidhmop 28969 The Hilbert space identity operator is a Hermitian operator. (Contributed by NM, 22-Apr-2006.) (New usage is discouraged.)
Iop ∈ HrmOp

Theorem0hmop 28970 The identically zero function is a Hermitian operator. (Contributed by NM, 8-Aug-2006.) (New usage is discouraged.)
0hop ∈ HrmOp

Theorem0lnop 28971 The identically zero function is a linear Hilbert space operator. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)
0hop ∈ LinOp

Theorem0lnfn 28972 The identically zero function is a linear Hilbert space functional. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
( ℋ × {0}) ∈ LinFn

Theoremnmop0 28973 The norm of the zero operator is zero. (Contributed by NM, 8-Feb-2006.) (New usage is discouraged.)
(normop‘ 0hop ) = 0

Theoremnmfn0 28974 The norm of the identically zero functional is zero. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.)
(normfn‘( ℋ × {0})) = 0

TheoremhmopbdoptHIL 28975 A Hermitian operator is a bounded linear operator (Hellinger-Toeplitz Theorem). (Contributed by NM, 18-Jan-2008.) (New usage is discouraged.)
(𝑇 ∈ HrmOp → 𝑇 ∈ BndLinOp)

Theoremhoddii 28976 Distributive law for Hilbert space operator difference. (Interestingly, the reverse distributive law hocsubdiri 28767 does not require linearity.) (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.)
𝑅 ∈ LinOp    &   𝑆: ℋ⟶ ℋ    &   𝑇: ℋ⟶ ℋ       (𝑅 ∘ (𝑆op 𝑇)) = ((𝑅𝑆) −op (𝑅𝑇))

Theoremhoddi 28977 Distributive law for Hilbert space operator difference. (Interestingly, the reverse distributive law hocsubdiri 28767 does not require linearity.) (Contributed by NM, 23-Aug-2006.) (New usage is discouraged.)
((𝑅 ∈ LinOp ∧ 𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (𝑅 ∘ (𝑆op 𝑇)) = ((𝑅𝑆) −op (𝑅𝑇)))

Theoremnmop0h 28978 The norm of any operator on the trivial Hilbert space is zero. (This is the reason we need ℋ ≠ 0 in nmopun 29001.) (Contributed by NM, 24-Feb-2006.) (New usage is discouraged.)
(( ℋ = 0𝑇: ℋ⟶ ℋ) → (normop𝑇) = 0)

Theoremidlnop 28979 The identity function (restricted to Hilbert space) is a linear operator. (Contributed by NM, 24-Jan-2006.) (New usage is discouraged.)
( I ↾ ℋ) ∈ LinOp

Theorem0bdop 28980 The identically zero operator is bounded. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
0hop ∈ BndLinOp

Theoremadj0 28981 Adjoint of the zero operator. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.)

Theoremnmlnop0iALT 28982 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       ((normop𝑇) = 0 ↔ 𝑇 = 0hop )

Theoremnmlnop0iHIL 28983 A linear operator with a zero norm is identically zero. (Contributed by NM, 18-Jan-2008.) (New usage is discouraged.)
𝑇 ∈ LinOp       ((normop𝑇) = 0 ↔ 𝑇 = 0hop )

Theoremnmlnopgt0i 28984 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       (𝑇 ≠ 0hop ↔ 0 < (normop𝑇))

Theoremnmlnop0 28985 A linear operator with a zero norm is identically zero. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)
(𝑇 ∈ LinOp → ((normop𝑇) = 0 ↔ 𝑇 = 0hop ))

Theoremnmlnopne0 28986 A linear operator with a nonzero norm is nonzero. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)
(𝑇 ∈ LinOp → ((normop𝑇) ≠ 0 ↔ 𝑇 ≠ 0hop ))

Theoremlnopmi 28987 The scalar product of a linear operator is a linear operator. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
𝑇 ∈ LinOp       (𝐴 ∈ ℂ → (𝐴 ·op 𝑇) ∈ LinOp)

Theoremlnophsi 28988 The sum of two linear operators is linear. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
𝑆 ∈ LinOp    &   𝑇 ∈ LinOp       (𝑆 +op 𝑇) ∈ LinOp

Theoremlnophdi 28989 The difference of two linear operators is linear. (Contributed by NM, 27-Jul-2006.) (New usage is discouraged.)
𝑆 ∈ LinOp    &   𝑇 ∈ LinOp       (𝑆op 𝑇) ∈ LinOp

Theoremlnopcoi 28990 The composition of two linear operators is linear. (Contributed by NM, 8-Mar-2006.) (New usage is discouraged.)
𝑆 ∈ LinOp    &   𝑇 ∈ LinOp       (𝑆𝑇) ∈ LinOp

Theoremlnopco0i 28991 The composition of a linear operator with one whose norm is zero. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
𝑆 ∈ LinOp    &   𝑇 ∈ LinOp       ((normop𝑇) = 0 → (normop‘(𝑆𝑇)) = 0)

Theoremlnopeq0lem1 28992 Lemma for lnopeq0i 28994. Apply the generalized polarization identity polid2i 28142 to the quadratic form ((𝑇𝑥), 𝑥). (Contributed by NM, 26-Jul-2006.) (New usage is discouraged.)
𝑇 ∈ LinOp    &   𝐴 ∈ ℋ    &   𝐵 ∈ ℋ       ((𝑇𝐴) ·ih 𝐵) = (((((𝑇‘(𝐴 + 𝐵)) ·ih (𝐴 + 𝐵)) − ((𝑇‘(𝐴 𝐵)) ·ih (𝐴 𝐵))) + (i · (((𝑇‘(𝐴 + (i · 𝐵))) ·ih (𝐴 + (i · 𝐵))) − ((𝑇‘(𝐴 (i · 𝐵))) ·ih (𝐴 (i · 𝐵)))))) / 4)

Theoremlnopeq0lem2 28993 Lemma for lnopeq0i 28994. (Contributed by NM, 26-Jul-2006.) (New usage is discouraged.)
𝑇 ∈ LinOp       ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝑇𝐴) ·ih 𝐵) = (((((𝑇‘(𝐴 + 𝐵)) ·ih (𝐴 + 𝐵)) − ((𝑇‘(𝐴 𝐵)) ·ih (𝐴 𝐵))) + (i · (((𝑇‘(𝐴 + (i · 𝐵))) ·ih (𝐴 + (i · 𝐵))) − ((𝑇‘(𝐴 (i · 𝐵))) ·ih (𝐴 (i · 𝐵)))))) / 4))

Theoremlnopeq0i 28994* A condition implying that a linear Hilbert space operator is identically zero. Unlike ho01i 28815 for arbitrary operators, when the operator is linear we need to consider only the values of the quadratic form (𝑇𝑥) ·ih 𝑥). (Contributed by NM, 26-Jul-2006.) (New usage is discouraged.)
𝑇 ∈ LinOp       (∀𝑥 ∈ ℋ ((𝑇𝑥) ·ih 𝑥) = 0 ↔ 𝑇 = 0hop )

Theoremlnopeqi 28995* 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    &   𝑈 ∈ LinOp       (∀𝑥 ∈ ℋ ((𝑇𝑥) ·ih 𝑥) = ((𝑈𝑥) ·ih 𝑥) ↔ 𝑇 = 𝑈)

Theoremlnopeq 28996* 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 ∧ 𝑈 ∈ LinOp) → (∀𝑥 ∈ ℋ ((𝑇𝑥) ·ih 𝑥) = ((𝑈𝑥) ·ih 𝑥) ↔ 𝑇 = 𝑈))

Theoremlnopunilem1 28997* Lemma for lnopunii 28999. (Contributed by NM, 14-May-2005.) (New usage is discouraged.)
𝑇 ∈ LinOp    &   𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥)    &   𝐴 ∈ ℋ    &   𝐵 ∈ ℋ    &   𝐶 ∈ ℂ       (ℜ‘(𝐶 · ((𝑇𝐴) ·ih (𝑇𝐵)))) = (ℜ‘(𝐶 · (𝐴 ·ih 𝐵)))

Theoremlnopunilem2 28998* Lemma for lnopunii 28999. (Contributed by NM, 12-May-2005.) (New usage is discouraged.)
𝑇 ∈ LinOp    &   𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥)    &   𝐴 ∈ ℋ    &   𝐵 ∈ ℋ       ((𝑇𝐴) ·ih (𝑇𝐵)) = (𝐴 ·ih 𝐵)

Theoremlnopunii 28999* If a linear operator (whose range is ) 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    &   𝑇: ℋ–onto→ ℋ    &   𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥)       𝑇 ∈ UniOp

Theoremelunop2 29000* 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–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥)))

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