P. 320 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 31901-32000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	lnopsubi 31901	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 31902	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 31903	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 31904	Move a scalar product out of a composition of operators. The operator 𝑇 must be linear, unlike homco1 31728 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 31905	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 31906	The identically zero function is a continuous Hilbert space operator. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)
⊢ 0hop ∈ ContOp
Theorem	0cnfn 31907	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 31908	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 31909	The Hilbert space identity operator is a Hermitian operator. (Contributed by NM, 22-Apr-2006.) (New usage is discouraged.)
⊢ Iop ∈ HrmOp
Theorem	0hmop 31910	The identically zero function is a Hermitian operator. (Contributed by NM, 8-Aug-2006.) (New usage is discouraged.)
⊢ 0hop ∈ HrmOp
Theorem	0lnop 31911	The identically zero function is a linear Hilbert space operator. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)
⊢ 0hop ∈ LinOp
Theorem	0lnfn 31912	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 31913	The norm of the zero operator is zero. (Contributed by NM, 8-Feb-2006.) (New usage is discouraged.)
⊢ (normop‘ 0hop ) = 0
Theorem	nmfn0 31914	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 31915	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 31916	Distributive law for Hilbert space operator difference. (Interestingly, the reverse distributive law hocsubdiri 31707 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 31917	Distributive law for Hilbert space operator difference. (Interestingly, the reverse distributive law hocsubdiri 31707 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 31918	The norm of any operator on the trivial Hilbert space is zero. (This is the reason we need ℋ ≠ 0ℋ in nmopun 31941.) (Contributed by NM, 24-Feb-2006.) (New usage is discouraged.)
⊢ (( ℋ = 0ℋ ∧ 𝑇: ℋ⟶ ℋ) → (normop‘𝑇) = 0)
Theorem	idlnop 31919	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 31920	The identically zero operator is bounded. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
⊢ 0hop ∈ BndLinOp
Theorem	adj0 31921	Adjoint of the zero operator. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.)
⊢ (adjℎ‘ 0hop ) = 0hop
Theorem	nmlnop0iALT 31922	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 31923	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 31924	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 31925	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 31926	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 31927	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 31928	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 31929	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 31930	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 31931	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 31932	Lemma for lnopeq0i 31934. Apply the generalized polarization identity polid2i 31084 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 31933	Lemma for lnopeq0i 31934. (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 31934*	A condition implying that a linear Hilbert space operator is identically zero. Unlike ho01i 31755 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 31935*	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 31936*	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 31937*	Lemma for lnopunii 31939. (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 31938*	Lemma for lnopunii 31939. (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 31939*	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 31940*	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 31941	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 31942	A unitary operator is a bounded linear operator. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
⊢ (𝑇 ∈ UniOp → 𝑇 ∈ BndLinOp)
Theorem	lnophmlem1 31943*	Lemma for lnophmi 31945. (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 31944*	Lemma for lnophmi 31945. (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 31945*	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 31946*	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 31947	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 31948	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 31949	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 31950	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 31951	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 31952	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 31953*	Lemma for nmcopexi 31954 and nmcfnexi 31978. 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 31954	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 31955	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 31956	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 31957	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 31958	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 31959	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 31960*	Lemma for lnopconi 31961 and lnfnconi 31982. (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 31961*	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 31962*	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 31963	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 31964	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 31965	A continuous linear operator is a bounded linear operator. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.)
⊢ (LinOp ∩ ContOp) = BndLinOp
Theorem	lnopcnre 31966	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))
Theorem	lnfnli 31967	Basic property of a linear Hilbert space 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	lnfnfi 31968	A linear Hilbert space functional is a functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)
⊢ 𝑇 ∈ LinFn    ⇒   ⊢ 𝑇: ℋ⟶ℂ
Theorem	lnfn0i 31969	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    ⇒   ⊢ (𝑇‘0ℎ) = 0
Theorem	lnfnaddi 31970	Additive property of a linear Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)
⊢ 𝑇 ∈ LinFn    ⇒   ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘(𝐴 +ℎ 𝐵)) = ((𝑇‘𝐴) + (𝑇‘𝐵)))
Theorem	lnfnmuli 31971	Multiplicative property of a linear Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)
⊢ 𝑇 ∈ LinFn    ⇒   ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (𝑇‘(𝐴 ·ℎ 𝐵)) = (𝐴 · (𝑇‘𝐵)))
Theorem	lnfnaddmuli 31972	Sum/product property of a linear Hilbert space functional. (Contributed by NM, 13-Feb-2006.) (New usage is discouraged.)
⊢ 𝑇 ∈ LinFn    ⇒   ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝑇‘(𝐵 +ℎ (𝐴 ·ℎ 𝐶))) = ((𝑇‘𝐵) + (𝐴 · (𝑇‘𝐶))))
Theorem	lnfnsubi 31973	Subtraction property for a linear Hilbert space functional. (Contributed by NM, 13-Feb-2006.) (New usage is discouraged.)
⊢ 𝑇 ∈ LinFn    ⇒   ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘(𝐴 −ℎ 𝐵)) = ((𝑇‘𝐴) − (𝑇‘𝐵)))
Theorem	lnfn0 31974	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 → (𝑇‘0ℎ) = 0)
Theorem	lnfnmul 31975	Multiplicative property of a linear Hilbert space functional. (Contributed by NM, 30-May-2006.) (New usage is discouraged.)
⊢ ((𝑇 ∈ LinFn ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (𝑇‘(𝐴 ·ℎ 𝐵)) = (𝐴 · (𝑇‘𝐵)))
Theorem	nmbdfnlbi 31976	A lower bound for the norm of a bounded linear functional. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.)
⊢ (𝑇 ∈ LinFn ∧ (normfn‘𝑇) ∈ ℝ)    ⇒   ⊢ (𝐴 ∈ ℋ → (abs‘(𝑇‘𝐴)) ≤ ((normfn‘𝑇) · (normℎ‘𝐴)))
Theorem	nmbdfnlb 31977	A lower bound for the norm of a bounded linear functional. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.)
⊢ ((𝑇 ∈ LinFn ∧ (normfn‘𝑇) ∈ ℝ ∧ 𝐴 ∈ ℋ) → (abs‘(𝑇‘𝐴)) ≤ ((normfn‘𝑇) · (normℎ‘𝐴)))
Theorem	nmcfnexi 31978	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    &   ⊢ 𝑇 ∈ ContFn    ⇒   ⊢ (normfn‘𝑇) ∈ ℝ
Theorem	nmcfnlbi 31979	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    &   ⊢ 𝑇 ∈ ContFn    ⇒   ⊢ (𝐴 ∈ ℋ → (abs‘(𝑇‘𝐴)) ≤ ((normfn‘𝑇) · (normℎ‘𝐴)))
Theorem	nmcfnex 31980	The norm of a continuous linear Hilbert space functional exists. Theorem 3.5(i) of [Beran] p. 99. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
⊢ ((𝑇 ∈ LinFn ∧ 𝑇 ∈ ContFn) → (normfn‘𝑇) ∈ ℝ)
Theorem	nmcfnlb 31981	A lower bound of the norm of a continuous linear Hilbert space functional. Theorem 3.5(ii) of [Beran] p. 99. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
⊢ ((𝑇 ∈ LinFn ∧ 𝑇 ∈ ContFn ∧ 𝐴 ∈ ℋ) → (abs‘(𝑇‘𝐴)) ≤ ((normfn‘𝑇) · (normℎ‘𝐴)))
Theorem	lnfnconi 31982*	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.) (Proof shortened by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.)
⊢ 𝑇 ∈ LinFn    ⇒   ⊢ (𝑇 ∈ ContFn ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (abs‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦)))
Theorem	lnfncon 31983*	A condition equivalent to "𝑇 is continuous" when 𝑇 is linear. Theorem 3.5(iii) of [Beran] p. 99. (Contributed by NM, 16-Feb-2006.) (New usage is discouraged.)
⊢ (𝑇 ∈ LinFn → (𝑇 ∈ ContFn ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (abs‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦))))
Theorem	lnfncnbd 31984	A linear functional is continuous iff it is bounded. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.)
⊢ (𝑇 ∈ LinFn → (𝑇 ∈ ContFn ↔ (normfn‘𝑇) ∈ ℝ))
Theorem	imaelshi 31985	The image of a subspace under a linear operator is a subspace. (Contributed by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
⊢ 𝑇 ∈ LinOp    &   ⊢ 𝐴 ∈ Sℋ    ⇒   ⊢ (𝑇 “ 𝐴) ∈ Sℋ
Theorem	rnelshi 31986	The range of a linear operator is a subspace. (Contributed by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.)
⊢ 𝑇 ∈ LinOp    ⇒   ⊢ ran 𝑇 ∈ Sℋ
Theorem	nlelshi 31987	The null space of a linear functional is a subspace. (Contributed by NM, 11-Feb-2006.) (Revised by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.)
⊢ 𝑇 ∈ LinFn    ⇒   ⊢ (null‘𝑇) ∈ Sℋ
Theorem	nlelchi 31988	The null space of a continuous linear functional is a closed subspace. Remark 3.8 of [Beran] p. 103. (Contributed by NM, 11-Feb-2006.) (Proof shortened by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
⊢ 𝑇 ∈ LinFn    &   ⊢ 𝑇 ∈ ContFn    ⇒   ⊢ (null‘𝑇) ∈ Cℋ
20.6.11  Riesz lemma
Theorem	riesz3i 31989*	A continuous linear functional can be expressed as an inner product. Existence part of Theorem 3.9 of [Beran] p. 104. (Contributed by NM, 13-Feb-2006.) (New usage is discouraged.)
⊢ 𝑇 ∈ LinFn    &   ⊢ 𝑇 ∈ ContFn    ⇒   ⊢ ∃𝑤 ∈ ℋ ∀𝑣 ∈ ℋ (𝑇‘𝑣) = (𝑣 ·ih 𝑤)
Theorem	riesz4i 31990*	A continuous linear functional can be expressed as an inner product. Uniqueness part of Theorem 3.9 of [Beran] p. 104. (Contributed by NM, 13-Feb-2006.) (New usage is discouraged.)
⊢ 𝑇 ∈ LinFn    &   ⊢ 𝑇 ∈ ContFn    ⇒   ⊢ ∃!𝑤 ∈ ℋ ∀𝑣 ∈ ℋ (𝑇‘𝑣) = (𝑣 ·ih 𝑤)
Theorem	riesz4 31991*	A continuous linear functional can be expressed as an inner product. Uniqueness part of Theorem 3.9 of [Beran] p. 104. See riesz2 31993 for the bounded linear functional version. (Contributed by NM, 16-Feb-2006.) (New usage is discouraged.)
⊢ (𝑇 ∈ (LinFn ∩ ContFn) → ∃!𝑤 ∈ ℋ ∀𝑣 ∈ ℋ (𝑇‘𝑣) = (𝑣 ·ih 𝑤))
Theorem	riesz1 31992*	Part 1 of the Riesz representation theorem for bounded linear functionals. A linear functional is bounded iff its value can be expressed as an inner product. Part of Theorem 17.3 of [Halmos] p. 31. For part 2, see riesz2 31993. For the continuous linear functional version, see riesz3i 31989 and riesz4 31991. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.)
⊢ (𝑇 ∈ LinFn → ((normfn‘𝑇) ∈ ℝ ↔ ∃𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑇‘𝑥) = (𝑥 ·ih 𝑦)))
Theorem	riesz2 31993*	Part 2 of the Riesz representation theorem for bounded linear functionals. The value of a bounded linear functional corresponds to a unique inner product. Part of Theorem 17.3 of [Halmos] p. 31. For part 1, see riesz1 31992. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.)
⊢ ((𝑇 ∈ LinFn ∧ (normfn‘𝑇) ∈ ℝ) → ∃!𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑇‘𝑥) = (𝑥 ·ih 𝑦))
20.6.12  Adjoints (cont.)
Theorem	cnlnadjlem1 31994*	Lemma for cnlnadji 32003 (Theorem 3.10 of [Beran] p. 104: every continuous linear operator has an adjoint). The value of the auxiliary functional 𝐺. (Contributed by NM, 16-Feb-2006.) (New usage is discouraged.)
⊢ 𝑇 ∈ LinOp    &   ⊢ 𝑇 ∈ ContOp    &   ⊢ 𝐺 = (𝑔 ∈ ℋ ↦ ((𝑇‘𝑔) ·ih 𝑦))    ⇒   ⊢ (𝐴 ∈ ℋ → (𝐺‘𝐴) = ((𝑇‘𝐴) ·ih 𝑦))
Theorem	cnlnadjlem2 31995*	Lemma for cnlnadji 32003. 𝐺 is a continuous linear functional. (Contributed by NM, 16-Feb-2006.) (New usage is discouraged.)
⊢ 𝑇 ∈ LinOp    &   ⊢ 𝑇 ∈ ContOp    &   ⊢ 𝐺 = (𝑔 ∈ ℋ ↦ ((𝑇‘𝑔) ·ih 𝑦))    ⇒   ⊢ (𝑦 ∈ ℋ → (𝐺 ∈ LinFn ∧ 𝐺 ∈ ContFn))
Theorem	cnlnadjlem3 31996*	Lemma for cnlnadji 32003. By riesz4 31991, 𝐵 is the unique vector such that (𝑇‘𝑣) ·ih 𝑦) = (𝑣 ·ih 𝑤) for all 𝑣. (Contributed by NM, 17-Feb-2006.) (New usage is discouraged.)
⊢ 𝑇 ∈ LinOp    &   ⊢ 𝑇 ∈ ContOp    &   ⊢ 𝐺 = (𝑔 ∈ ℋ ↦ ((𝑇‘𝑔) ·ih 𝑦))    &   ⊢ 𝐵 = (℩𝑤 ∈ ℋ ∀𝑣 ∈ ℋ ((𝑇‘𝑣) ·ih 𝑦) = (𝑣 ·ih 𝑤))    ⇒   ⊢ (𝑦 ∈ ℋ → 𝐵 ∈ ℋ)
Theorem	cnlnadjlem4 31997*	Lemma for cnlnadji 32003. The values of auxiliary function 𝐹 are vectors. (Contributed by NM, 17-Feb-2006.) (Proof shortened by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.)
⊢ 𝑇 ∈ LinOp    &   ⊢ 𝑇 ∈ ContOp    &   ⊢ 𝐺 = (𝑔 ∈ ℋ ↦ ((𝑇‘𝑔) ·ih 𝑦))    &   ⊢ 𝐵 = (℩𝑤 ∈ ℋ ∀𝑣 ∈ ℋ ((𝑇‘𝑣) ·ih 𝑦) = (𝑣 ·ih 𝑤))    &   ⊢ 𝐹 = (𝑦 ∈ ℋ ↦ 𝐵)    ⇒   ⊢ (𝐴 ∈ ℋ → (𝐹‘𝐴) ∈ ℋ)
Theorem	cnlnadjlem5 31998*	Lemma for cnlnadji 32003. 𝐹 is an adjoint of 𝑇 (later, we will show it is unique). (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.)
⊢ 𝑇 ∈ LinOp    &   ⊢ 𝑇 ∈ ContOp    &   ⊢ 𝐺 = (𝑔 ∈ ℋ ↦ ((𝑇‘𝑔) ·ih 𝑦))    &   ⊢ 𝐵 = (℩𝑤 ∈ ℋ ∀𝑣 ∈ ℋ ((𝑇‘𝑣) ·ih 𝑦) = (𝑣 ·ih 𝑤))    &   ⊢ 𝐹 = (𝑦 ∈ ℋ ↦ 𝐵)    ⇒   ⊢ ((𝐴 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝑇‘𝐶) ·ih 𝐴) = (𝐶 ·ih (𝐹‘𝐴)))
Theorem	cnlnadjlem6 31999*	Lemma for cnlnadji 32003. 𝐹 is linear. (Contributed by NM, 17-Feb-2006.) (New usage is discouraged.)
⊢ 𝑇 ∈ LinOp    &   ⊢ 𝑇 ∈ ContOp    &   ⊢ 𝐺 = (𝑔 ∈ ℋ ↦ ((𝑇‘𝑔) ·ih 𝑦))    &   ⊢ 𝐵 = (℩𝑤 ∈ ℋ ∀𝑣 ∈ ℋ ((𝑇‘𝑣) ·ih 𝑦) = (𝑣 ·ih 𝑤))    &   ⊢ 𝐹 = (𝑦 ∈ ℋ ↦ 𝐵)    ⇒   ⊢ 𝐹 ∈ LinOp
Theorem	cnlnadjlem7 32000*	Lemma for cnlnadji 32003. Helper lemma to show that 𝐹 is continuous. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.)
⊢ 𝑇 ∈ LinOp    &   ⊢ 𝑇 ∈ ContOp    &   ⊢ 𝐺 = (𝑔 ∈ ℋ ↦ ((𝑇‘𝑔) ·ih 𝑦))    &   ⊢ 𝐵 = (℩𝑤 ∈ ℋ ∀𝑣 ∈ ℋ ((𝑇‘𝑣) ·ih 𝑦) = (𝑣 ·ih 𝑤))    &   ⊢ 𝐹 = (𝑦 ∈ ℋ ↦ 𝐵)    ⇒   ⊢ (𝐴 ∈ ℋ → (normℎ‘(𝐹‘𝐴)) ≤ ((normop‘𝑇) · (normℎ‘𝐴)))