P. 315 of Theorem List - Metamath Proof Explorer

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**Theorem List for Metamath Proof Explorer -** 31401-31500 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	nmfnrepnf 31401	The norm of a Hilbert space functional is either real or plus infinity. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.)
⊢ (𝑇: ℋ⟶ℂ → ((norm_fn‘𝑇) ∈ ℝ ↔ (norm_fn‘𝑇) ≠ +∞))

Theorem	nlfnval 31402	Value of the null space of a Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)
⊢ (𝑇: ℋ⟶ℂ → (null‘𝑇) = (^◡𝑇 “ {0}))

Theorem	elcnfn 31403*	Property defining a continuous functional. (Contributed by NM, 11-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
⊢ (𝑇 ∈ ContFn ↔ (𝑇: ℋ⟶ℂ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℝ⁺ ∃𝑧 ∈ ℝ⁺ ∀𝑤 ∈ ℋ ((norm_ℎ‘(𝑤 −_ℎ 𝑥)) < 𝑧 → (abs‘((𝑇‘𝑤) − (𝑇‘𝑥))) < 𝑦)))

Theorem	ellnfn 31404*	Property defining a linear functional. (Contributed by NM, 11-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
⊢ (𝑇 ∈ LinFn ↔ (𝑇: ℋ⟶ℂ ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) = ((𝑥 · (𝑇‘𝑦)) + (𝑇‘𝑧))))

Theorem	lnfnf 31405	A linear Hilbert space functional is a functional. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.)
⊢ (𝑇 ∈ LinFn → 𝑇: ℋ⟶ℂ)

Theorem	dfadj2 31406*	Alternate definition of the adjoint of a Hilbert space operator. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.)
⊢ adj_ℎ = {⟨𝑡, 𝑢⟩ ∣ (𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑡‘𝑦)) = ((𝑢‘𝑥) ·_ih 𝑦))}

Theorem	funadj 31407	Functionality of the adjoint function. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)
⊢ Fun adj_ℎ

Theorem	dmadjss 31408	The domain of the adjoint function is a subset of the maps from ℋ to ℋ. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)
⊢ dom adj_ℎ ⊆ ( ℋ ↑_m ℋ)

Theorem	dmadjop 31409	A member of the domain of the adjoint function is a Hilbert space operator. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)
⊢ (𝑇 ∈ dom adj_ℎ → 𝑇: ℋ⟶ ℋ)

Theorem	adjeu 31410*	Elementhood in the domain of the adjoint function. (Contributed by Mario Carneiro, 11-Sep-2015.) (Revised by Mario Carneiro, 24-Dec-2016.) (New usage is discouraged.)
⊢ (𝑇: ℋ⟶ ℋ → (𝑇 ∈ dom adj_ℎ ↔ ∃!𝑢 ∈ ( ℋ ↑_m ℋ)∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑢‘𝑥) ·_ih 𝑦)))

Theorem	adjval 31411*	Value of the adjoint function for 𝑇 in the domain of adj_ℎ. (Contributed by NM, 19-Feb-2006.) (Revised by Mario Carneiro, 24-Dec-2016.) (New usage is discouraged.)
⊢ (𝑇 ∈ dom adj_ℎ → (adj_ℎ‘𝑇) = (℩𝑢 ∈ ( ℋ ↑_m ℋ)∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑢‘𝑥) ·_ih 𝑦)))

Theorem	adjval2 31412*	Value of the adjoint function. (Contributed by NM, 19-Feb-2006.) (New usage is discouraged.)
⊢ (𝑇 ∈ dom adj_ℎ → (adj_ℎ‘𝑇) = (℩𝑢 ∈ ( ℋ ↑_m ℋ)∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·_ih 𝑦) = (𝑥 ·_ih (𝑢‘𝑦))))

Theorem	cnvadj 31413	The adjoint function equals its converse. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)
⊢ ^◡adj_ℎ = adj_ℎ

Theorem	funcnvadj 31414	The converse of the adjoint function is a function. (Contributed by NM, 25-Jan-2006.) (New usage is discouraged.)
⊢ Fun ^◡adj_ℎ

Theorem	adj1o 31415	The adjoint function maps one-to-one onto its domain. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)
⊢ adj_ℎ:dom adj_ℎ–1-1-onto→dom adj_ℎ

Theorem	dmadjrn 31416	The adjoint of an operator belongs to the adjoint function's domain. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)
⊢ (𝑇 ∈ dom adj_ℎ → (adj_ℎ‘𝑇) ∈ dom adj_ℎ)

Theorem	eigvecval 31417*	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 → (eigvec‘𝑇) = {𝑥 ∈ ( ℋ ∖ 0_ℋ) ∣ ∃𝑦 ∈ ℂ (𝑇‘𝑥) = (𝑦 ·_ℎ 𝑥)})

Theorem	eigvalfval 31418*	The eigenvalues of eigenvectors of a Hilbert space operator. (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.)
⊢ (𝑇: ℋ⟶ ℋ → (eigval‘𝑇) = (𝑥 ∈ (eigvec‘𝑇) ↦ (((𝑇‘𝑥) ·_ih 𝑥) / ((norm_ℎ‘𝑥)↑2))))

Theorem	specval 31419*	The value of the spectrum of an operator. (Contributed by NM, 11-Apr-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
⊢ (𝑇: ℋ⟶ ℋ → (Lambda‘𝑇) = {𝑥 ∈ ℂ ∣ ¬ (𝑇 −_op (𝑥 ·_op ( I ↾ ℋ))): ℋ–1-1→ ℋ})

Theorem	speccl 31420	The spectrum of an operator is a set of complex numbers. (Contributed by NM, 11-Apr-2006.) (New usage is discouraged.)
⊢ (𝑇: ℋ⟶ ℋ → (Lambda‘𝑇) ⊆ ℂ)

Theorem	hhlnoi 31421	The linear operators of Hilbert space. (Contributed by NM, 19-Nov-2007.) (Revised by Mario Carneiro, 19-Nov-2013.) (New usage is discouraged.)
⊢ 𝑈 = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ & ⊢ 𝐿 = (𝑈 LnOp 𝑈) ⇒ ⊢ LinOp = 𝐿

Theorem	hhnmoi 31422	The norm of an operator in Hilbert space. (Contributed by NM, 19-Nov-2007.) (Revised by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.)
⊢ 𝑈 = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ & ⊢ 𝑁 = (𝑈 normOp_OLD 𝑈) ⇒ ⊢ norm_op = 𝑁

Theorem	hhbloi 31423	A bounded linear operator in Hilbert space. (Contributed by NM, 19-Nov-2007.) (Revised by Mario Carneiro, 19-Nov-2013.) (New usage is discouraged.)
⊢ 𝑈 = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ & ⊢ 𝐵 = (𝑈 BLnOp 𝑈) ⇒ ⊢ BndLinOp = 𝐵

Theorem	hh0oi 31424	The zero operator in Hilbert space. (Contributed by NM, 7-Dec-2007.) (New usage is discouraged.)
⊢ 𝑈 = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ & ⊢ 𝑍 = (𝑈 0_op 𝑈) ⇒ ⊢ 0_hop = 𝑍

Theorem	hhcno 31425	The continuous operators of Hilbert space. (Contributed by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
⊢ 𝐷 = (norm_ℎ ∘ −_ℎ ) & ⊢ 𝐽 = (MetOpen‘𝐷) ⇒ ⊢ ContOp = (𝐽 Cn 𝐽)

Theorem	hhcnf 31426	The continuous functionals of Hilbert space. (Contributed by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
⊢ 𝐷 = (norm_ℎ ∘ −_ℎ ) & ⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ 𝐾 = (TopOpen‘ℂ_fld) ⇒ ⊢ ContFn = (𝐽 Cn 𝐾)

Theorem	dmadjrnb 31427	The adjoint of an operator belongs to the adjoint function's domain. (Note: the converse is dependent on our definition of function value, since it uses ndmfv 6926.) (Contributed by NM, 19-Feb-2006.) (New usage is discouraged.)
⊢ (𝑇 ∈ dom adj_ℎ ↔ (adj_ℎ‘𝑇) ∈ dom adj_ℎ)

Theorem	nmoplb 31428	A lower bound for an operator norm. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)
⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℋ ∧ (norm_ℎ‘𝐴) ≤ 1) → (norm_ℎ‘(𝑇‘𝐴)) ≤ (norm_op‘𝑇))

Theorem	nmopub 31429*	An upper bound for an operator norm. (Contributed by NM, 7-Mar-2006.) (New usage is discouraged.)
⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℝ^*) → ((norm_op‘𝑇) ≤ 𝐴 ↔ ∀𝑥 ∈ ℋ ((norm_ℎ‘𝑥) ≤ 1 → (norm_ℎ‘(𝑇‘𝑥)) ≤ 𝐴)))

Theorem	nmopub2tALT 31430*	An upper bound for an operator norm. (Contributed by NM, 12-Apr-2006.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ ((𝑇: ℋ⟶ ℋ ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ ∀𝑥 ∈ ℋ (norm_ℎ‘(𝑇‘𝑥)) ≤ (𝐴 · (norm_ℎ‘𝑥))) → (norm_op‘𝑇) ≤ 𝐴)

Theorem	nmopub2tHIL 31431*	An upper bound for an operator norm. (Contributed by NM, 13-Dec-2007.) (New usage is discouraged.)
⊢ ((𝑇: ℋ⟶ ℋ ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ ∀𝑥 ∈ ℋ (norm_ℎ‘(𝑇‘𝑥)) ≤ (𝐴 · (norm_ℎ‘𝑥))) → (norm_op‘𝑇) ≤ 𝐴)

Theorem	nmopge0 31432	The norm of any Hilbert space operator is nonnegative. (Contributed by NM, 9-Feb-2006.) (New usage is discouraged.)
⊢ (𝑇: ℋ⟶ ℋ → 0 ≤ (norm_op‘𝑇))

Theorem	nmopgt0 31433	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: ℋ⟶ ℋ → ((norm_op‘𝑇) ≠ 0 ↔ 0 < (norm_op‘𝑇)))

Theorem	cnopc 31434*	Basic continuity property of a continuous Hilbert space operator. (Contributed by NM, 2-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
⊢ ((𝑇 ∈ ContOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℝ⁺) → ∃𝑥 ∈ ℝ⁺ ∀𝑦 ∈ ℋ ((norm_ℎ‘(𝑦 −_ℎ 𝐴)) < 𝑥 → (norm_ℎ‘((𝑇‘𝑦) −_ℎ (𝑇‘𝐴))) < 𝐵))

Theorem	lnopl 31435	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 ·_ℎ 𝐵) +_ℎ 𝐶)) = ((𝐴 ·_ℎ (𝑇‘𝐵)) +_ℎ (𝑇‘𝐶)))

Theorem	unop 31436	Basic inner product property of a unitary operator. (Contributed by NM, 22-Jan-2006.) (New usage is discouraged.)
⊢ ((𝑇 ∈ UniOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝑇‘𝐴) ·_ih (𝑇‘𝐵)) = (𝐴 ·_ih 𝐵))

Theorem	unopf1o 31437	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)

Theorem	unopnorm 31438	A unitary operator is idempotent in the norm. (Contributed by NM, 25-Feb-2006.) (New usage is discouraged.)
⊢ ((𝑇 ∈ UniOp ∧ 𝐴 ∈ ℋ) → (norm_ℎ‘(𝑇‘𝐴)) = (norm_ℎ‘𝐴))

Theorem	cnvunop 31439	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)

Theorem	unopadj 31440	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) → ((𝑇‘𝐴) ·_ih 𝐵) = (𝐴 ·_ih (^◡𝑇‘𝐵)))

Theorem	unoplin 31441	A unitary operator is linear. Theorem in [AkhiezerGlazman] p. 72. (Contributed by NM, 22-Jan-2006.) (New usage is discouraged.)
⊢ (𝑇 ∈ UniOp → 𝑇 ∈ LinOp)

Theorem	counop 31442	The composition of two unitary operators is unitary. (Contributed by NM, 22-Jan-2006.) (New usage is discouraged.)
⊢ ((𝑆 ∈ UniOp ∧ 𝑇 ∈ UniOp) → (𝑆 ∘ 𝑇) ∈ UniOp)

Theorem	hmop 31443	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 (𝑇‘𝐵)) = ((𝑇‘𝐴) ·_ih 𝐵))

Theorem	hmopre 31444	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) → ((𝑇‘𝐴) ·_ih 𝐴) ∈ ℝ)

Theorem	nmfnlb 31445	A lower bound for a functional norm. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
⊢ ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ ∧ (norm_ℎ‘𝐴) ≤ 1) → (abs‘(𝑇‘𝐴)) ≤ (norm_fn‘𝑇))

Theorem	nmfnleub 31446*	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^*) → ((norm_fn‘𝑇) ≤ 𝐴 ↔ ∀𝑥 ∈ ℋ ((norm_ℎ‘𝑥) ≤ 1 → (abs‘(𝑇‘𝑥)) ≤ 𝐴)))

Theorem	nmfnleub2 31447*	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‘(𝑇‘𝑥)) ≤ (𝐴 · (norm_ℎ‘𝑥))) → (norm_fn‘𝑇) ≤ 𝐴)

Theorem	nmfnge0 31448	The norm of any Hilbert space functional is nonnegative. (Contributed by NM, 24-May-2006.) (New usage is discouraged.)
⊢ (𝑇: ℋ⟶ℂ → 0 ≤ (norm_fn‘𝑇))

Theorem	elnlfn 31449	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 ∧ (𝑇‘𝐴) = 0)))

Theorem	elnlfn2 31450	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)) → (𝑇‘𝐴) = 0)

Theorem	cnfnc 31451*	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_ℎ‘(𝑦 −_ℎ 𝐴)) < 𝑥 → (abs‘((𝑇‘𝑦) − (𝑇‘𝐴))) < 𝐵))

Theorem	lnfnl 31452	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 ·_ℎ 𝐵) +_ℎ 𝐶)) = ((𝐴 · (𝑇‘𝐵)) + (𝑇‘𝐶)))

Theorem	adjcl 31453	Closure of the adjoint of a Hilbert space operator. (Contributed by NM, 17-Jun-2006.) (New usage is discouraged.)
⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝐴 ∈ ℋ) → ((adj_ℎ‘𝑇)‘𝐴) ∈ ℋ)

Theorem	adj1 31454	Property of an adjoint Hilbert space operator. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)
⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·_ih (𝑇‘𝐵)) = (((adj_ℎ‘𝑇)‘𝐴) ·_ih 𝐵))

Theorem	adj2 31455	Property of an adjoint Hilbert space operator. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)
⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝑇‘𝐴) ·_ih 𝐵) = (𝐴 ·_ih ((adj_ℎ‘𝑇)‘𝐵)))

Theorem	adjeq 31456*	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 ((𝑇‘𝑥) ·_ih 𝑦) = (𝑥 ·_ih (𝑆‘𝑦))) → (adj_ℎ‘𝑇) = 𝑆)

Theorem	adjadj 31457	Double adjoint. Theorem 3.11(iv) of [Beran] p. 106. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)
⊢ (𝑇 ∈ dom adj_ℎ → (adj_ℎ‘(adj_ℎ‘𝑇)) = 𝑇)

Theorem	adjvalval 31458*	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 adj_ℎ ∧ 𝐴 ∈ ℋ) → ((adj_ℎ‘𝑇)‘𝐴) = (℩𝑤 ∈ ℋ ∀𝑥 ∈ ℋ ((𝑇‘𝑥) ·_ih 𝐴) = (𝑥 ·_ih 𝑤)))

Theorem	unopadj2 31459	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 → (adj_ℎ‘𝑇) = ^◡𝑇)

Theorem	hmopadj 31460	A Hermitian operator is self-adjoint. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)
⊢ (𝑇 ∈ HrmOp → (adj_ℎ‘𝑇) = 𝑇)

Theorem	hmdmadj 31461	Every Hermitian operator has an adjoint. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)
⊢ (𝑇 ∈ HrmOp → 𝑇 ∈ dom adj_ℎ)

Theorem	hmopadj2 31462	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_ℎ → (𝑇 ∈ HrmOp ↔ (adj_ℎ‘𝑇) = 𝑇))

Theorem	hmoplin 31463	A Hermitian operator is linear. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)
⊢ (𝑇 ∈ HrmOp → 𝑇 ∈ LinOp)

Theorem	brafval 31464*	The bra of a vector, expressed as ⟨𝐴 ∣ in Dirac notation. See df-bra 31371. (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)))

Theorem	braval 31465	A bra-ket juxtaposition, expressed as ⟨𝐴 ∣ 𝐵⟩ in Dirac notation, equals the inner product of the vectors. Based on definition of bra in [Prugovecki] p. 186. (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))

Theorem	braadd 31466	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 +_ℎ 𝐶)) = (((bra‘𝐴)‘𝐵) + ((bra‘𝐴)‘𝐶)))

Theorem	bramul 31467	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 ·_ℎ 𝐶)) = (𝐵 · ((bra‘𝐴)‘𝐶)))

Theorem	brafn 31468	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)

Theorem	bralnfn 31469	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)

Theorem	bracl 31470	Closure of the bra function. (Contributed by NM, 23-May-2006.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((bra‘𝐴)‘𝐵) ∈ ℂ)

Theorem	bra0 31471	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) = ( ℋ × {0})

Theorem	brafnmul 31472	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 ·_ℎ 𝐵)) = ((∗‘𝐴) ·_fn (bra‘𝐵)))

Theorem	kbfval 31473*	The outer product of two vectors, expressed as ∣ 𝐴⟩⟨𝐵 ∣ in Dirac notation. See df-kb 31372. (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) ·_ℎ 𝐴)))

Theorem	kbop 31474	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)

Theorem	kbval 31475	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) ·_ℎ 𝐴))

Theorem	kbmul 31476	Multiplication property of outer product. (Contributed by NM, 31-May-2006.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ·_ℎ 𝐵) ketbra 𝐶) = (𝐵 ketbra ((∗‘𝐴) ·_ℎ 𝐶)))

Theorem	kbpj 31477	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 ∧ (norm_ℎ‘𝐴) = 1) → (𝐴 ketbra 𝐴) = (proj_ℎ‘(span‘{𝐴})))

Theorem	eleigvec 31478*	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_ℎ ∧ ∃𝑥 ∈ ℂ (𝑇‘𝐴) = (𝑥 ·_ℎ 𝐴))))

Theorem	eleigvec2 31479	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_ℎ ∧ (𝑇‘𝐴) ∈ (span‘{𝐴}))))

Theorem	eleigveccl 31480	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)

Theorem	eigvalval 31481	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) = (((𝑇‘𝐴) ·_ih 𝐴) / ((norm_ℎ‘𝐴)↑2)))

Theorem	eigvalcl 31482	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)

Theorem	eigvec1 31483	Property of an eigenvector. (Contributed by NM, 12-Mar-2006.) (New usage is discouraged.)
⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ (eigvec‘𝑇)) → ((𝑇‘𝐴) = (((eigval‘𝑇)‘𝐴) ·_ℎ 𝐴) ∧ 𝐴 ≠ 0_ℎ))

Theorem	eighmre 31484	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)

Theorem	eighmorth 31485	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)

Theorem	nmopnegi 31486	Value of the norm of the negative of a Hilbert space operator. Unlike nmophmi 31552, the operator does not have to be bounded. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
⊢ 𝑇: ℋ⟶ ℋ ⇒ ⊢ (norm_op‘(-1 ·_op 𝑇)) = (norm_op‘𝑇)

Theorem	lnop0 31487	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_ℎ) = 0_ℎ)

Theorem	lnopmul 31488	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 ·_ℎ 𝐵)) = (𝐴 ·_ℎ (𝑇‘𝐵)))

Theorem	lnopli 31489	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 ·_ℎ 𝐵) +_ℎ 𝐶)) = ((𝐴 ·_ℎ (𝑇‘𝐵)) +_ℎ (𝑇‘𝐶)))

Theorem	lnopfi 31490	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

Theorem	lnop0i 31491	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_ℎ) = 0_ℎ

Theorem	lnopaddi 31492	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 +_ℎ 𝐵)) = ((𝑇‘𝐴) +_ℎ (𝑇‘𝐵)))

Theorem	lnopmuli 31493	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 ·_ℎ 𝐵)) = (𝐴 ·_ℎ (𝑇‘𝐵)))

Theorem	lnopaddmuli 31494	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 +_ℎ (𝐴 ·_ℎ 𝐶))) = ((𝑇‘𝐵) +_ℎ (𝐴 ·_ℎ (𝑇‘𝐶))))

Theorem	lnopsubi 31495	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 −_ℎ 𝐵)) = ((𝑇‘𝐴) −_ℎ (𝑇‘𝐵)))

Theorem	lnopsubmuli 31496	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 −_ℎ (𝐴 ·_ℎ 𝐶))) = ((𝑇‘𝐵) −_ℎ (𝐴 ·_ℎ (𝑇‘𝐶))))

Theorem	lnopmulsubi 31497	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 ·_ℎ 𝐵) −_ℎ 𝐶)) = ((𝐴 ·_ℎ (𝑇‘𝐵)) −_ℎ (𝑇‘𝐶)))

Theorem	homco2 31498	Move a scalar product out of a composition of operators. The operator 𝑇 must be linear, unlike homco1 31322 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 (𝑇 ∘ 𝑈)))

Theorem	idunop 31499	The identity function (restricted to Hilbert space) is a unitary operator. (Contributed by NM, 21-Jan-2006.) (New usage is discouraged.)
⊢ ( I ↾ ℋ) ∈ UniOp

Theorem	0cnop 31500	The identically zero function is a continuous Hilbert space operator. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)
⊢ 0_hop ∈ ContOp

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45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45800 459 45801-45900 460 45901-46000 461 46001-46100 462 46101-46200 463 46201-46300 464 46301-46400 465 46401-46500 466 46501-46600 467 46601-46700 468 46701-46800 469 46801-46900 470 46901-47000 471 47001-47100 472 47101-47200 473 47201-47300 474 47301-47400 475 47401-47500 476 47501-47600 477 47601-47700 478 47701-47800 479 47801-47900 480 47901-47940

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