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Theorem List for Metamath Proof Explorer - 28801-28900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlnopmuli 28801 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 28802 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 28803 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 28804 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 28805 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 28806 Move a scalar product out of a composition of operators. The operator 𝑇 must be linear, unlike homco1 28630 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 28807 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 28808 The identically zero function is a continuous Hilbert space operator. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)
0hop ∈ ContOp

Theorem0cnfn 28809 The identically zero function is a continuous Hilbert space functional. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)
( ℋ × {0}) ∈ ContFn

Theoremidcnop 28810 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 28811 The Hilbert space identity operator is a Hermitian operator. (Contributed by NM, 22-Apr-2006.) (New usage is discouraged.)
Iop ∈ HrmOp

Theorem0hmop 28812 The identically zero function is a Hermitian operator. (Contributed by NM, 8-Aug-2006.) (New usage is discouraged.)
0hop ∈ HrmOp

Theorem0lnop 28813 The identically zero function is a linear Hilbert space operator. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)
0hop ∈ LinOp

Theorem0lnfn 28814 The identically zero function is a linear Hilbert space functional. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
( ℋ × {0}) ∈ LinFn

Theoremnmop0 28815 The norm of the zero operator is zero. (Contributed by NM, 8-Feb-2006.) (New usage is discouraged.)
(normop‘ 0hop ) = 0

Theoremnmfn0 28816 The norm of the identically zero functional is zero. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.)
(normfn‘( ℋ × {0})) = 0

TheoremhmopbdoptHIL 28817 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 28818 Distributive law for Hilbert space operator difference. (Interestingly, the reverse distributive law hocsubdiri 28609 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 28819 Distributive law for Hilbert space operator difference. (Interestingly, the reverse distributive law hocsubdiri 28609 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 28820 The norm of any operator on the trivial Hilbert space is zero. (This is the reason we need ℋ ≠ 0 in nmopun 28843.) (Contributed by NM, 24-Feb-2006.) (New usage is discouraged.)
(( ℋ = 0𝑇: ℋ⟶ ℋ) → (normop𝑇) = 0)

Theoremidlnop 28821 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 28822 The identically zero operator is bounded. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
0hop ∈ BndLinOp

Theoremadj0 28823 Adjoint of the zero operator. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.)

Theoremnmlnop0iALT 28824 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 28825 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 28826 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 28827 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 28828 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 28829 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 28830 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 28831 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 28832 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 28833 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 28834 Lemma for lnopeq0i 28836. Apply the generalized polarization identity polid2i 27984 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 28835 Lemma for lnopeq0i 28836. (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 28836* A condition implying that a linear Hilbert space operator is identically zero. Unlike ho01i 28657 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 28837* 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 28838* 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 28839* Lemma for lnopunii 28841. (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 28840* Lemma for lnopunii 28841. (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 28841* 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 28842* 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)))

Theoremnmopun 28843 Norm of a unitary Hilbert space operator. (Contributed by NM, 25-Feb-2006.) (New usage is discouraged.)
(( ℋ ≠ 0𝑇 ∈ UniOp) → (normop𝑇) = 1)

Theoremunopbd 28844 A unitary operator is a bounded linear operator. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
(𝑇 ∈ UniOp → 𝑇 ∈ BndLinOp)

Theoremlnophmlem1 28845* Lemma for lnophmi 28847. (Contributed by NM, 24-Jan-2006.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ    &   𝑇 ∈ LinOp    &   𝑥 ∈ ℋ (𝑥 ·ih (𝑇𝑥)) ∈ ℝ       (𝐴 ·ih (𝑇𝐴)) ∈ ℝ

Theoremlnophmlem2 28846* Lemma for lnophmi 28847. (Contributed by NM, 24-Jan-2006.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ    &   𝑇 ∈ LinOp    &   𝑥 ∈ ℋ (𝑥 ·ih (𝑇𝑥)) ∈ ℝ       (𝐴 ·ih (𝑇𝐵)) = ((𝑇𝐴) ·ih 𝐵)

Theoremlnophmi 28847* A linear operator is Hermitian if 𝑥 ·ih (𝑇𝑥) 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 (𝑇𝑥)) ∈ ℝ       𝑇 ∈ HrmOp

Theoremlnophm 28848* A linear operator is Hermitian if 𝑥 ·ih (𝑇𝑥) 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 (𝑇𝑥)) ∈ ℝ) → 𝑇 ∈ HrmOp)

Theoremhmops 28849 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)

Theoremhmopm 28850 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)

Theoremhmopd 28851 The difference of two Hermitian operators is Hermitian. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → (𝑇op 𝑈) ∈ HrmOp)

Theoremhmopco 28852 The composition of two commuting Hermitian operators is Hermitian. (Contributed by NM, 22-Aug-2006.) (New usage is discouraged.)
((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp ∧ (𝑇𝑈) = (𝑈𝑇)) → (𝑇𝑈) ∈ HrmOp)

Theoremnmbdoplbi 28853 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 → (norm‘(𝑇𝐴)) ≤ ((normop𝑇) · (norm𝐴)))

Theoremnmbdoplb 28854 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) → (norm‘(𝑇𝐴)) ≤ ((normop𝑇) · (norm𝐴)))

Theoremnmcexi 28855* Lemma for nmcopexi 28856 and nmcfnexi 28880. 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 ((norm𝑧) < 𝑦 → (𝑁‘(𝑇𝑧)) < 1)    &   (𝑆𝑇) = sup({𝑚 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥)))}, ℝ*, < )    &   (𝑥 ∈ ℋ → (𝑁‘(𝑇𝑥)) ∈ ℝ)    &   (𝑁‘(𝑇‘0)) = 0    &   (((𝑦 / 2) ∈ ℝ+𝑥 ∈ ℋ) → ((𝑦 / 2) · (𝑁‘(𝑇𝑥))) = (𝑁‘(𝑇‘((𝑦 / 2) · 𝑥))))       (𝑆𝑇) ∈ ℝ

Theoremnmcopexi 28856 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       (normop𝑇) ∈ ℝ

Theoremnmcoplbi 28857 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 → (norm‘(𝑇𝐴)) ≤ ((normop𝑇) · (norm𝐴)))

Theoremnmcopex 28858 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) → (normop𝑇) ∈ ℝ)

Theoremnmcoplb 28859 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) → (norm‘(𝑇𝐴)) ≤ ((normop𝑇) · (norm𝐴)))

Theoremnmophmi 28860 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) · (normop𝑇)))

Theorembdophmi 28861 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)

Theoremlnconi 28862* Lemma for lnopconi 28863 and lnfnconi 28884. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)
(𝑇𝐶𝑆 ∈ ℝ)    &   ((𝑇𝐶𝑦 ∈ ℋ) → (𝑁‘(𝑇𝑦)) ≤ (𝑆 · (norm𝑦)))    &   (𝑇𝐶 ↔ ∀𝑥 ∈ ℋ ∀𝑧 ∈ ℝ+𝑦 ∈ ℝ+𝑤 ∈ ℋ ((norm‘(𝑤 𝑥)) < 𝑦 → (𝑁‘((𝑇𝑤)𝑀(𝑇𝑥))) < 𝑧))    &   (𝑦 ∈ ℋ → (𝑁‘(𝑇𝑦)) ∈ ℝ)    &   ((𝑤 ∈ ℋ ∧ 𝑥 ∈ ℋ) → (𝑇‘(𝑤 𝑥)) = ((𝑇𝑤)𝑀(𝑇𝑥)))       (𝑇𝐶 ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑥 · (norm𝑦)))

Theoremlnopconi 28863* 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 (norm‘(𝑇𝑦)) ≤ (𝑥 · (norm𝑦)))

Theoremlnopcon 28864* 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 (norm‘(𝑇𝑦)) ≤ (𝑥 · (norm𝑦))))

Theoremlnopcnbd 28865 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))

Theoremlncnopbd 28866 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 28867 A continuous linear operator is a bounded linear operator. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.)
(LinOp ∩ ContOp) = BndLinOp

Theoremlnopcnre 28868 A linear operator is continuous iff it is bounded. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
(𝑇 ∈ LinOp → (𝑇 ∈ ContOp ↔ (normop𝑇) ∈ ℝ))

Theoremlnfnli 28869 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 · 𝐵) + 𝐶)) = ((𝐴 · (𝑇𝐵)) + (𝑇𝐶)))

Theoremlnfnfi 28870 A linear Hilbert space functional is a functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)
𝑇 ∈ LinFn       𝑇: ℋ⟶ℂ

Theoremlnfn0i 28871 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‘0) = 0

Theoremlnfnaddi 28872 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 + 𝐵)) = ((𝑇𝐴) + (𝑇𝐵)))

Theoremlnfnmuli 28873 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 · 𝐵)) = (𝐴 · (𝑇𝐵)))

Theoremlnfnaddmuli 28874 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 + (𝐴 · 𝐶))) = ((𝑇𝐵) + (𝐴 · (𝑇𝐶))))

Theoremlnfnsubi 28875 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 𝐵)) = ((𝑇𝐴) − (𝑇𝐵)))

Theoremlnfn0 28876 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‘0) = 0)

Theoremlnfnmul 28877 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 · 𝐵)) = (𝐴 · (𝑇𝐵)))

Theoremnmbdfnlbi 28878 A lower bound for the norm of a bounded linear functional. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.)
(𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ)       (𝐴 ∈ ℋ → (abs‘(𝑇𝐴)) ≤ ((normfn𝑇) · (norm𝐴)))

Theoremnmbdfnlb 28879 A lower bound for the norm of a bounded linear functional. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.)
((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ ∧ 𝐴 ∈ ℋ) → (abs‘(𝑇𝐴)) ≤ ((normfn𝑇) · (norm𝐴)))

Theoremnmcfnexi 28880 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       (normfn𝑇) ∈ ℝ

Theoremnmcfnlbi 28881 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‘(𝑇𝐴)) ≤ ((normfn𝑇) · (norm𝐴)))

Theoremnmcfnex 28882 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) → (normfn𝑇) ∈ ℝ)

Theoremnmcfnlb 28883 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‘(𝑇𝐴)) ≤ ((normfn𝑇) · (norm𝐴)))

Theoremlnfnconi 28884* 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‘(𝑇𝑦)) ≤ (𝑥 · (norm𝑦)))

Theoremlnfncon 28885* 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‘(𝑇𝑦)) ≤ (𝑥 · (norm𝑦))))

Theoremlnfncnbd 28886 A linear functional is continuous iff it is bounded. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.)
(𝑇 ∈ LinFn → (𝑇 ∈ ContFn ↔ (normfn𝑇) ∈ ℝ))

Theoremimaelshi 28887 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    &   𝐴S       (𝑇𝐴) ∈ S

Theoremrnelshi 28888 The range of a linear operator is a subspace. (Contributed by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.)
𝑇 ∈ LinOp       ran 𝑇S

Theoremnlelshi 28889 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) ∈ S

Theoremnlelchi 28890 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) ∈ C

19.6.11  Riesz lemma

Theoremriesz3i 28891* 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 (𝑇𝑣) = (𝑣 ·ih 𝑤)

Theoremriesz4i 28892* 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 (𝑇𝑣) = (𝑣 ·ih 𝑤)

Theoremriesz4 28893* A continuous linear functional can be expressed as an inner product. Uniqueness part of Theorem 3.9 of [Beran] p. 104. See riesz2 28895 for the bounded linear functional version. (Contributed by NM, 16-Feb-2006.) (New usage is discouraged.)
(𝑇 ∈ (LinFn ∩ ContFn) → ∃!𝑤 ∈ ℋ ∀𝑣 ∈ ℋ (𝑇𝑣) = (𝑣 ·ih 𝑤))

Theoremriesz1 28894* 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 28895. For the continuous linear functional version, see riesz3i 28891 and riesz4 28893. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.)
(𝑇 ∈ LinFn → ((normfn𝑇) ∈ ℝ ↔ ∃𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑇𝑥) = (𝑥 ·ih 𝑦)))

Theoremriesz2 28895* 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 28894. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.)
((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ) → ∃!𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑇𝑥) = (𝑥 ·ih 𝑦))

Theoremcnlnadjlem1 28896* Lemma for cnlnadji 28905 (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 ↦ ((𝑇𝑔) ·ih 𝑦))       (𝐴 ∈ ℋ → (𝐺𝐴) = ((𝑇𝐴) ·ih 𝑦))

Theoremcnlnadjlem2 28897* Lemma for cnlnadji 28905. 𝐺 is a continuous linear functional. (Contributed by NM, 16-Feb-2006.) (New usage is discouraged.)
𝑇 ∈ LinOp    &   𝑇 ∈ ContOp    &   𝐺 = (𝑔 ∈ ℋ ↦ ((𝑇𝑔) ·ih 𝑦))       (𝑦 ∈ ℋ → (𝐺 ∈ LinFn ∧ 𝐺 ∈ ContFn))

Theoremcnlnadjlem3 28898* Lemma for cnlnadji 28905. By riesz4 28893, 𝐵 is the unique vector such that (𝑇𝑣) ·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 ↦ ((𝑇𝑔) ·ih 𝑦))    &   𝐵 = (𝑤 ∈ ℋ ∀𝑣 ∈ ℋ ((𝑇𝑣) ·ih 𝑦) = (𝑣 ·ih 𝑤))       (𝑦 ∈ ℋ → 𝐵 ∈ ℋ)

Theoremcnlnadjlem4 28899* Lemma for cnlnadji 28905. 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 ↦ ((𝑇𝑔) ·ih 𝑦))    &   𝐵 = (𝑤 ∈ ℋ ∀𝑣 ∈ ℋ ((𝑇𝑣) ·ih 𝑦) = (𝑣 ·ih 𝑤))    &   𝐹 = (𝑦 ∈ ℋ ↦ 𝐵)       (𝐴 ∈ ℋ → (𝐹𝐴) ∈ ℋ)

Theoremcnlnadjlem5 28900* Lemma for cnlnadji 28905. 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 ↦ ((𝑇𝑔) ·ih 𝑦))    &   𝐵 = (𝑤 ∈ ℋ ∀𝑣 ∈ ℋ ((𝑇𝑣) ·ih 𝑦) = (𝑣 ·ih 𝑤))    &   𝐹 = (𝑦 ∈ ℋ ↦ 𝐵)       ((𝐴 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝑇𝐶) ·ih 𝐴) = (𝐶 ·ih (𝐹𝐴)))

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