P. 323 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 32201-32300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	hmdmadj 32201	Every Hermitian operator has an adjoint. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)
⊢ (𝑇 ∈ HrmOp → 𝑇 ∈ dom adjℎ)
Theorem	hmopadj2 32202	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 adjℎ → (𝑇 ∈ HrmOp ↔ (adjℎ‘𝑇) = 𝑇))
Theorem	hmoplin 32203	A Hermitian operator is linear. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)
⊢ (𝑇 ∈ HrmOp → 𝑇 ∈ LinOp)
Theorem	brafval 32204*	The bra of a vector, expressed as ⟨𝐴 ∣ in Dirac notation. See df-bra 32111. (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 32205	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 32206	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 32207	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 32208	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 32209	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 32210	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 32211	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‘0ℎ) = ( ℋ × {0})
Theorem	brafnmul 32212	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 32213*	The outer product of two vectors, expressed as ∣ 𝐴⟩⟨𝐵 ∣ in Dirac notation. See df-kb 32112. (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 32214	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 32215	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 32216	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 32217	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 ∧ (normℎ‘𝐴) = 1) → (𝐴 ketbra 𝐴) = (projℎ‘(span‘{𝐴})))
Theorem	eleigvec 32218*	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 ≠ 0ℎ ∧ ∃𝑥 ∈ ℂ (𝑇‘𝐴) = (𝑥 ·ℎ 𝐴))))
Theorem	eleigvec2 32219	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 ≠ 0ℎ ∧ (𝑇‘𝐴) ∈ (span‘{𝐴}))))
Theorem	eleigveccl 32220	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 32221	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) / ((normℎ‘𝐴)↑2)))
Theorem	eigvalcl 32222	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 32223	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 ≠ 0ℎ))
Theorem	eighmre 32224	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 32225	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 32226	Value of the norm of the negative of a Hilbert space operator. Unlike nmophmi 32292, 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)) = (normop‘𝑇)
Theorem	lnop0 32227	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‘0ℎ) = 0ℎ)
Theorem	lnopmul 32228	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 32229	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 32230	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 32231	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‘0ℎ) = 0ℎ
Theorem	lnopaddi 32232	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 32233	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 32234	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 32235	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 32236	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 32237	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 32238	Move a scalar product out of a composition of operators. The operator 𝑇 must be linear, unlike homco1 32062 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 32239	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 32240	The identically zero function is a continuous Hilbert space operator. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)
⊢ 0hop ∈ ContOp
Theorem	0cnfn 32241	The identically zero function is a continuous Hilbert space functional. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)
⊢ ( ℋ × {0}) ∈ ContFn
Theorem	idcnop 32242	The identity function (restricted to Hilbert space) is a continuous operator. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)
⊢ ( I ↾ ℋ) ∈ ContOp
Theorem	idhmop 32243	The Hilbert space identity operator is a Hermitian operator. (Contributed by NM, 22-Apr-2006.) (New usage is discouraged.)
⊢ Iop ∈ HrmOp
Theorem	0hmop 32244	The identically zero function is a Hermitian operator. (Contributed by NM, 8-Aug-2006.) (New usage is discouraged.)
⊢ 0hop ∈ HrmOp
Theorem	0lnop 32245	The identically zero function is a linear Hilbert space operator. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)
⊢ 0hop ∈ LinOp
Theorem	0lnfn 32246	The identically zero function is a linear Hilbert space functional. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
⊢ ( ℋ × {0}) ∈ LinFn
Theorem	nmop0 32247	The norm of the zero operator is zero. (Contributed by NM, 8-Feb-2006.) (New usage is discouraged.)
⊢ (normop‘ 0hop ) = 0
Theorem	nmfn0 32248	The norm of the identically zero functional is zero. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.)
⊢ (normfn‘( ℋ × {0})) = 0
Theorem	hmopbdoptHIL 32249	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)
Theorem	hoddii 32250	Distributive law for Hilbert space operator difference. (Interestingly, the reverse distributive law hocsubdiri 32041 does not require linearity.) (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.)
⊢ 𝑅 ∈ LinOp    &   ⊢ 𝑆: ℋ⟶ ℋ    &   ⊢ 𝑇: ℋ⟶ ℋ    ⇒   ⊢ (𝑅 ∘ (𝑆 −op 𝑇)) = ((𝑅 ∘ 𝑆) −op (𝑅 ∘ 𝑇))
Theorem	hoddi 32251	Distributive law for Hilbert space operator difference. (Interestingly, the reverse distributive law hocsubdiri 32041 does not require linearity.) (Contributed by NM, 23-Aug-2006.) (New usage is discouraged.)
⊢ ((𝑅 ∈ LinOp ∧ 𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (𝑅 ∘ (𝑆 −op 𝑇)) = ((𝑅 ∘ 𝑆) −op (𝑅 ∘ 𝑇)))
Theorem	nmop0h 32252	The norm of any operator on the trivial Hilbert space is zero. (This is the reason we need ℋ ≠ 0ℋ in nmopun 32275.) (Contributed by NM, 24-Feb-2006.) (New usage is discouraged.)
⊢ (( ℋ = 0ℋ ∧ 𝑇: ℋ⟶ ℋ) → (normop‘𝑇) = 0)
Theorem	idlnop 32253	The identity function (restricted to Hilbert space) is a linear operator. (Contributed by NM, 24-Jan-2006.) (New usage is discouraged.)
⊢ ( I ↾ ℋ) ∈ LinOp
Theorem	0bdop 32254	The identically zero operator is bounded. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
⊢ 0hop ∈ BndLinOp
Theorem	adj0 32255	Adjoint of the zero operator. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.)
⊢ (adjℎ‘ 0hop ) = 0hop
Theorem	nmlnop0iALT 32256	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    ⇒   ⊢ ((normop‘𝑇) = 0 ↔ 𝑇 = 0hop )
Theorem	nmlnop0iHIL 32257	A linear operator with a zero norm is identically zero. (Contributed by NM, 18-Jan-2008.) (New usage is discouraged.)
⊢ 𝑇 ∈ LinOp    ⇒   ⊢ ((normop‘𝑇) = 0 ↔ 𝑇 = 0hop )
Theorem	nmlnopgt0i 32258	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 < (normop‘𝑇))
Theorem	nmlnop0 32259	A linear operator with a zero norm is identically zero. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)
⊢ (𝑇 ∈ LinOp → ((normop‘𝑇) = 0 ↔ 𝑇 = 0hop ))
Theorem	nmlnopne0 32260	A linear operator with a nonzero norm is nonzero. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)
⊢ (𝑇 ∈ LinOp → ((normop‘𝑇) ≠ 0 ↔ 𝑇 ≠ 0hop ))
Theorem	lnopmi 32261	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)
Theorem	lnophsi 32262	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
Theorem	lnophdi 32263	The difference of two linear operators is linear. (Contributed by NM, 27-Jul-2006.) (New usage is discouraged.)
⊢ 𝑆 ∈ LinOp    &   ⊢ 𝑇 ∈ LinOp    ⇒   ⊢ (𝑆 −op 𝑇) ∈ LinOp
Theorem	lnopcoi 32264	The composition of two linear operators is linear. (Contributed by NM, 8-Mar-2006.) (New usage is discouraged.)
⊢ 𝑆 ∈ LinOp    &   ⊢ 𝑇 ∈ LinOp    ⇒   ⊢ (𝑆 ∘ 𝑇) ∈ LinOp
Theorem	lnopco0i 32265	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    ⇒   ⊢ ((normop‘𝑇) = 0 → (normop‘(𝑆 ∘ 𝑇)) = 0)
Theorem	lnopeq0lem1 32266	Lemma for lnopeq0i 32268. Apply the generalized polarization identity polid2i 31418 to the quadratic form ((𝑇‘𝑥), 𝑥). (Contributed by NM, 26-Jul-2006.) (New usage is discouraged.)
⊢ 𝑇 ∈ LinOp    &   ⊢ 𝐴 ∈ ℋ    &   ⊢ 𝐵 ∈ ℋ    ⇒   ⊢ ((𝑇‘𝐴) ·ih 𝐵) = (((((𝑇‘(𝐴 +ℎ 𝐵)) ·ih (𝐴 +ℎ 𝐵)) − ((𝑇‘(𝐴 −ℎ 𝐵)) ·ih (𝐴 −ℎ 𝐵))) + (i · (((𝑇‘(𝐴 +ℎ (i ·ℎ 𝐵))) ·ih (𝐴 +ℎ (i ·ℎ 𝐵))) − ((𝑇‘(𝐴 −ℎ (i ·ℎ 𝐵))) ·ih (𝐴 −ℎ (i ·ℎ 𝐵)))))) / 4)
Theorem	lnopeq0lem2 32267	Lemma for lnopeq0i 32268. (Contributed by NM, 26-Jul-2006.) (New usage is discouraged.)
⊢ 𝑇 ∈ LinOp    ⇒   ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝑇‘𝐴) ·ih 𝐵) = (((((𝑇‘(𝐴 +ℎ 𝐵)) ·ih (𝐴 +ℎ 𝐵)) − ((𝑇‘(𝐴 −ℎ 𝐵)) ·ih (𝐴 −ℎ 𝐵))) + (i · (((𝑇‘(𝐴 +ℎ (i ·ℎ 𝐵))) ·ih (𝐴 +ℎ (i ·ℎ 𝐵))) − ((𝑇‘(𝐴 −ℎ (i ·ℎ 𝐵))) ·ih (𝐴 −ℎ (i ·ℎ 𝐵)))))) / 4))
Theorem	lnopeq0i 32268*	A condition implying that a linear Hilbert space operator is identically zero. Unlike ho01i 32089 for arbitrary operators, when the operator is linear we need to consider only the values of the quadratic form (𝑇‘𝑥) ·ih 𝑥). (Contributed by NM, 26-Jul-2006.) (New usage is discouraged.)
⊢ 𝑇 ∈ LinOp    ⇒   ⊢ (∀𝑥 ∈ ℋ ((𝑇‘𝑥) ·ih 𝑥) = 0 ↔ 𝑇 = 0hop )
Theorem	lnopeqi 32269*	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 ((𝑇‘𝑥) ·ih 𝑥) = ((𝑈‘𝑥) ·ih 𝑥) ↔ 𝑇 = 𝑈)
Theorem	lnopeq 32270*	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 ((𝑇‘𝑥) ·ih 𝑥) = ((𝑈‘𝑥) ·ih 𝑥) ↔ 𝑇 = 𝑈))
Theorem	lnopunilem1 32271*	Lemma for lnopunii 32273. (Contributed by NM, 14-May-2005.) (New usage is discouraged.)
⊢ 𝑇 ∈ LinOp    &   ⊢ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥)    &   ⊢ 𝐴 ∈ ℋ    &   ⊢ 𝐵 ∈ ℋ    &   ⊢ 𝐶 ∈ ℂ    ⇒   ⊢ (ℜ‘(𝐶 · ((𝑇‘𝐴) ·ih (𝑇‘𝐵)))) = (ℜ‘(𝐶 · (𝐴 ·ih 𝐵)))
Theorem	lnopunilem2 32272*	Lemma for lnopunii 32273. (Contributed by NM, 12-May-2005.) (New usage is discouraged.)
⊢ 𝑇 ∈ LinOp    &   ⊢ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥)    &   ⊢ 𝐴 ∈ ℋ    &   ⊢ 𝐵 ∈ ℋ    ⇒   ⊢ ((𝑇‘𝐴) ·ih (𝑇‘𝐵)) = (𝐴 ·ih 𝐵)
Theorem	lnopunii 32273*	If a linear operator (whose range is ℋ) 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 (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥)    ⇒   ⊢ 𝑇 ∈ UniOp
Theorem	elunop2 32274*	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 (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥)))
Theorem	nmopun 32275	Norm of a unitary Hilbert space operator. (Contributed by NM, 25-Feb-2006.) (New usage is discouraged.)
⊢ (( ℋ ≠ 0ℋ ∧ 𝑇 ∈ UniOp) → (normop‘𝑇) = 1)
Theorem	unopbd 32276	A unitary operator is a bounded linear operator. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
⊢ (𝑇 ∈ UniOp → 𝑇 ∈ BndLinOp)
Theorem	lnophmlem1 32277*	Lemma for lnophmi 32279. (Contributed by NM, 24-Jan-2006.) (New usage is discouraged.)
⊢ 𝐴 ∈ ℋ    &   ⊢ 𝐵 ∈ ℋ    &   ⊢ 𝑇 ∈ LinOp    &   ⊢ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑇‘𝑥)) ∈ ℝ    ⇒   ⊢ (𝐴 ·ih (𝑇‘𝐴)) ∈ ℝ
Theorem	lnophmlem2 32278*	Lemma for lnophmi 32279. (Contributed by NM, 24-Jan-2006.) (New usage is discouraged.)
⊢ 𝐴 ∈ ℋ    &   ⊢ 𝐵 ∈ ℋ    &   ⊢ 𝑇 ∈ LinOp    &   ⊢ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑇‘𝑥)) ∈ ℝ    ⇒   ⊢ (𝐴 ·ih (𝑇‘𝐵)) = ((𝑇‘𝐴) ·ih 𝐵)
Theorem	lnophmi 32279*	A linear operator is Hermitian if 𝑥 ·ih (𝑇‘𝑥) takes only real values. Remark in [ReedSimon] p. 195. (Contributed by NM, 24-Jan-2006.) (New usage is discouraged.)
⊢ 𝑇 ∈ LinOp    &   ⊢ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑇‘𝑥)) ∈ ℝ    ⇒   ⊢ 𝑇 ∈ HrmOp
Theorem	lnophm 32280*	A linear operator is Hermitian if 𝑥 ·ih (𝑇‘𝑥) takes only real values. Remark in [ReedSimon] p. 195. (Contributed by NM, 24-Jan-2006.) (New usage is discouraged.)
⊢ ((𝑇 ∈ LinOp ∧ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑇‘𝑥)) ∈ ℝ) → 𝑇 ∈ HrmOp)
Theorem	hmops 32281	The sum of two Hermitian operators is Hermitian. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
⊢ ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → (𝑇 +op 𝑈) ∈ HrmOp)
Theorem	hmopm 32282	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) → (𝐴 ·op 𝑇) ∈ HrmOp)
Theorem	hmopd 32283	The difference of two Hermitian operators is Hermitian. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
⊢ ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → (𝑇 −op 𝑈) ∈ HrmOp)
Theorem	hmopco 32284	The composition of two commuting Hermitian operators is Hermitian. (Contributed by NM, 22-Aug-2006.) (New usage is discouraged.)
⊢ ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp ∧ (𝑇 ∘ 𝑈) = (𝑈 ∘ 𝑇)) → (𝑇 ∘ 𝑈) ∈ HrmOp)
Theorem	nmbdoplbi 32285	A lower bound for the norm of a bounded linear operator. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
⊢ 𝑇 ∈ BndLinOp    ⇒   ⊢ (𝐴 ∈ ℋ → (normℎ‘(𝑇‘𝐴)) ≤ ((normop‘𝑇) · (normℎ‘𝐴)))
Theorem	nmbdoplb 32286	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 ∈ ℋ) → (normℎ‘(𝑇‘𝐴)) ≤ ((normop‘𝑇) · (normℎ‘𝐴)))
Theorem	nmcexi 32287*	Lemma for nmcopexi 32288 and nmcfnexi 32312. 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.)
⊢ ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ ℋ ((normℎ‘𝑧) < 𝑦 → (𝑁‘(𝑇‘𝑧)) < 1)    &   ⊢ (𝑆‘𝑇) = sup({𝑚 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇‘𝑥)))}, ℝ*, < )    &   ⊢ (𝑥 ∈ ℋ → (𝑁‘(𝑇‘𝑥)) ∈ ℝ)    &   ⊢ (𝑁‘(𝑇‘0ℎ)) = 0    &   ⊢ (((𝑦 / 2) ∈ ℝ+ ∧ 𝑥 ∈ ℋ) → ((𝑦 / 2) · (𝑁‘(𝑇‘𝑥))) = (𝑁‘(𝑇‘((𝑦 / 2) ·ℎ 𝑥))))    ⇒   ⊢ (𝑆‘𝑇) ∈ ℝ
Theorem	nmcopexi 32288	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    &   ⊢ 𝑇 ∈ ContOp    ⇒   ⊢ (normop‘𝑇) ∈ ℝ
Theorem	nmcoplbi 32289	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    &   ⊢ 𝑇 ∈ ContOp    ⇒   ⊢ (𝐴 ∈ ℋ → (normℎ‘(𝑇‘𝐴)) ≤ ((normop‘𝑇) · (normℎ‘𝐴)))
Theorem	nmcopex 32290	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) → (normop‘𝑇) ∈ ℝ)
Theorem	nmcoplb 32291	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 ∈ ℋ) → (normℎ‘(𝑇‘𝐴)) ≤ ((normop‘𝑇) · (normℎ‘𝐴)))
Theorem	nmophmi 32292	The norm of the scalar product of a bounded linear operator. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
⊢ 𝑇 ∈ BndLinOp    ⇒   ⊢ (𝐴 ∈ ℂ → (normop‘(𝐴 ·op 𝑇)) = ((abs‘𝐴) · (normop‘𝑇)))
Theorem	bdophmi 32293	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    ⇒   ⊢ (𝐴 ∈ ℂ → (𝐴 ·op 𝑇) ∈ BndLinOp)
Theorem	lnconi 32294*	Lemma for lnopconi 32295 and lnfnconi 32316. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)
⊢ (𝑇 ∈ 𝐶 → 𝑆 ∈ ℝ)    &   ⊢ ((𝑇 ∈ 𝐶 ∧ 𝑦 ∈ ℋ) → (𝑁‘(𝑇‘𝑦)) ≤ (𝑆 · (normℎ‘𝑦)))    &   ⊢ (𝑇 ∈ 𝐶 ↔ ∀𝑥 ∈ ℋ ∀𝑧 ∈ ℝ+ ∃𝑦 ∈ ℝ+ ∀𝑤 ∈ ℋ ((normℎ‘(𝑤 −ℎ 𝑥)) < 𝑦 → (𝑁‘((𝑇‘𝑤)𝑀(𝑇‘𝑥))) < 𝑧))    &   ⊢ (𝑦 ∈ ℋ → (𝑁‘(𝑇‘𝑦)) ∈ ℝ)    &   ⊢ ((𝑤 ∈ ℋ ∧ 𝑥 ∈ ℋ) → (𝑇‘(𝑤 −ℎ 𝑥)) = ((𝑇‘𝑤)𝑀(𝑇‘𝑥)))    ⇒   ⊢ (𝑇 ∈ 𝐶 ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦)))
Theorem	lnopconi 32295*	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    ⇒   ⊢ (𝑇 ∈ ContOp ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (normℎ‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦)))
Theorem	lnopcon 32296*	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 → (𝑇 ∈ ContOp ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (normℎ‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦))))
Theorem	lnopcnbd 32297	A linear operator is continuous iff it is bounded. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
⊢ (𝑇 ∈ LinOp → (𝑇 ∈ ContOp ↔ 𝑇 ∈ BndLinOp))
Theorem	lncnopbd 32298	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)
Theorem	lncnbd 32299	A continuous linear operator is a bounded linear operator. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.)
⊢ (LinOp ∩ ContOp) = BndLinOp
Theorem	lnopcnre 32300	A linear operator is continuous iff it is bounded. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
⊢ (𝑇 ∈ LinOp → (𝑇 ∈ ContOp ↔ (normop‘𝑇) ∈ ℝ))