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Theorem List for Metamath Proof Explorer - 29201-29300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremhodidi 29201 Difference of a Hilbert space operator from itself. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
𝑇: ℋ⟶ ℋ       (𝑇op 𝑇) = 0hop

Theoremho0coi 29202 Composition of the zero operator and a Hilbert space operator. (Contributed by NM, 9-Aug-2006.) (New usage is discouraged.)
𝑇: ℋ⟶ ℋ       ( 0hop𝑇) = 0hop

Theoremhoid1i 29203 Composition of Hilbert space operator with unit identity. (Contributed by NM, 15-Nov-2000.) (New usage is discouraged.)
𝑇: ℋ⟶ ℋ       (𝑇 ∘ Iop ) = 𝑇

Theoremhoid1ri 29204 Composition of Hilbert space operator with unit identity. (Contributed by NM, 15-Nov-2000.) (New usage is discouraged.)
𝑇: ℋ⟶ ℋ       ( Iop𝑇) = 𝑇

Theoremhoaddid1 29205 Sum of a Hilbert space operator with the zero operator. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.)
(𝑇: ℋ⟶ ℋ → (𝑇 +op 0hop ) = 𝑇)

Theoremhodid 29206 Difference of a Hilbert space operator from itself. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
(𝑇: ℋ⟶ ℋ → (𝑇op 𝑇) = 0hop )

Theoremhon0 29207 A Hilbert space operator is not empty. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)
(𝑇: ℋ⟶ ℋ → ¬ 𝑇 = ∅)

Theoremhodseqi 29208 Subtraction and addition of equal Hilbert space operators. (Contributed by NM, 27-Aug-2004.) (New usage is discouraged.)
𝑆: ℋ⟶ ℋ    &   𝑇: ℋ⟶ ℋ       (𝑆 +op (𝑇op 𝑆)) = 𝑇

Theoremho0subi 29209 Subtraction of Hilbert space operators expressed in terms of difference from zero. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
𝑆: ℋ⟶ ℋ    &   𝑇: ℋ⟶ ℋ       (𝑆op 𝑇) = (𝑆 +op ( 0hopop 𝑇))

Theoremhonegsubi 29210 Relationship between Hilbert operator addition and subtraction. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
𝑆: ℋ⟶ ℋ    &   𝑇: ℋ⟶ ℋ       (𝑆 +op (-1 ·op 𝑇)) = (𝑆op 𝑇)

Theoremho0sub 29211 Subtraction of Hilbert space operators expressed in terms of difference from zero. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.)
((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (𝑆op 𝑇) = (𝑆 +op ( 0hopop 𝑇)))

Theoremhosubid1 29212 The zero operator subtracted from a Hilbert space operator. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.)
(𝑇: ℋ⟶ ℋ → (𝑇op 0hop ) = 𝑇)

Theoremhonegsub 29213 Relationship between Hilbert space operator addition and subtraction. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
((𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (𝑇 +op (-1 ·op 𝑈)) = (𝑇op 𝑈))

Theoremhomulid2 29214 An operator equals its scalar product with one. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)
(𝑇: ℋ⟶ ℋ → (1 ·op 𝑇) = 𝑇)

Theoremhomco1 29215 Associative law for scalar product and composition of operators. (Contributed by NM, 13-Aug-2006.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → ((𝐴 ·op 𝑇) ∘ 𝑈) = (𝐴 ·op (𝑇𝑈)))

Theoremhomulass 29216 Scalar product associative law for Hilbert space operators. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → ((𝐴 · 𝐵) ·op 𝑇) = (𝐴 ·op (𝐵 ·op 𝑇)))

Theoremhoadddi 29217 Scalar product distributive law for Hilbert space operators. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (𝐴 ·op (𝑇 +op 𝑈)) = ((𝐴 ·op 𝑇) +op (𝐴 ·op 𝑈)))

Theoremhoadddir 29218 Scalar product reverse distributive law for Hilbert space operators. (Contributed by NM, 25-Aug-2006.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → ((𝐴 + 𝐵) ·op 𝑇) = ((𝐴 ·op 𝑇) +op (𝐵 ·op 𝑇)))

Theoremhomul12 29219 Swap first and second factors in a nested operator scalar product. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝐴 ·op (𝐵 ·op 𝑇)) = (𝐵 ·op (𝐴 ·op 𝑇)))

Theoremhonegneg 29220 Double negative of a Hilbert space operator. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
(𝑇: ℋ⟶ ℋ → (-1 ·op (-1 ·op 𝑇)) = 𝑇)

Theoremhosubneg 29221 Relationship between operator subtraction and negative. (Contributed by NM, 25-Aug-2006.) (New usage is discouraged.)
((𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (𝑇op (-1 ·op 𝑈)) = (𝑇 +op 𝑈))

Theoremhosubdi 29222 Scalar product distributive law for operator difference. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (𝐴 ·op (𝑇op 𝑈)) = ((𝐴 ·op 𝑇) −op (𝐴 ·op 𝑈)))

Theoremhonegdi 29223 Distribution of negative over addition. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
((𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (-1 ·op (𝑇 +op 𝑈)) = ((-1 ·op 𝑇) +op (-1 ·op 𝑈)))

Theoremhonegsubdi 29224 Distribution of negative over subtraction. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
((𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (-1 ·op (𝑇op 𝑈)) = ((-1 ·op 𝑇) +op 𝑈))

Theoremhonegsubdi2 29225 Distribution of negative over subtraction. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
((𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (-1 ·op (𝑇op 𝑈)) = (𝑈op 𝑇))

Theoremhosubsub2 29226 Law for double subtraction of Hilbert space operators. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (𝑆op (𝑇op 𝑈)) = (𝑆 +op (𝑈op 𝑇)))

Theoremhosub4 29227 Rearrangement of 4 terms in a mixed addition and subtraction of Hilbert space operators. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
(((𝑅: ℋ⟶ ℋ ∧ 𝑆: ℋ⟶ ℋ) ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ)) → ((𝑅 +op 𝑆) −op (𝑇 +op 𝑈)) = ((𝑅op 𝑇) +op (𝑆op 𝑈)))

Theoremhosubadd4 29228 Rearrangement of 4 terms in a mixed addition and subtraction of Hilbert space operators. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
(((𝑅: ℋ⟶ ℋ ∧ 𝑆: ℋ⟶ ℋ) ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ)) → ((𝑅op 𝑆) −op (𝑇op 𝑈)) = ((𝑅 +op 𝑈) −op (𝑆 +op 𝑇)))

Theoremhoaddsubass 29229 Associative-type law for addition and subtraction of Hilbert space operators. (Contributed by NM, 25-Aug-2006.) (New usage is discouraged.)
((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → ((𝑆 +op 𝑇) −op 𝑈) = (𝑆 +op (𝑇op 𝑈)))

Theoremhoaddsub 29230 Law for operator addition and subtraction of Hilbert space operators. (Contributed by NM, 25-Aug-2006.) (New usage is discouraged.)
((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → ((𝑆 +op 𝑇) −op 𝑈) = ((𝑆op 𝑈) +op 𝑇))

Theoremhosubsub 29231 Law for double subtraction of Hilbert space operators. (Contributed by NM, 25-Aug-2006.) (New usage is discouraged.)
((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (𝑆op (𝑇op 𝑈)) = ((𝑆op 𝑇) +op 𝑈))

Theoremhosubsub4 29232 Law for double subtraction of Hilbert space operators. (Contributed by NM, 25-Aug-2006.) (New usage is discouraged.)
((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → ((𝑆op 𝑇) −op 𝑈) = (𝑆op (𝑇 +op 𝑈)))

Theoremho2times 29233 Two times a Hilbert space operator. (Contributed by NM, 26-Aug-2006.) (New usage is discouraged.)
(𝑇: ℋ⟶ ℋ → (2 ·op 𝑇) = (𝑇 +op 𝑇))

Theoremhoaddsubassi 29234 Associativity of sum and difference of Hilbert space operators. (Contributed by NM, 27-Aug-2004.) (New usage is discouraged.)
𝑅: ℋ⟶ ℋ    &   𝑆: ℋ⟶ ℋ    &   𝑇: ℋ⟶ ℋ       ((𝑅 +op 𝑆) −op 𝑇) = (𝑅 +op (𝑆op 𝑇))

Theoremhoaddsubi 29235 Law for sum and difference of Hilbert space operators. (Contributed by NM, 27-Aug-2004.) (New usage is discouraged.)
𝑅: ℋ⟶ ℋ    &   𝑆: ℋ⟶ ℋ    &   𝑇: ℋ⟶ ℋ       ((𝑅 +op 𝑆) −op 𝑇) = ((𝑅op 𝑇) +op 𝑆)

Theoremhosd1i 29236 Hilbert space operator sum expressed in terms of difference. (Contributed by NM, 27-Aug-2004.) (New usage is discouraged.)
𝑇: ℋ⟶ ℋ    &   𝑈: ℋ⟶ ℋ       (𝑇 +op 𝑈) = (𝑇op ( 0hopop 𝑈))

Theoremhosd2i 29237 Hilbert space operator sum expressed in terms of difference. (Contributed by NM, 27-Aug-2004.) (New usage is discouraged.)
𝑇: ℋ⟶ ℋ    &   𝑈: ℋ⟶ ℋ       (𝑇 +op 𝑈) = (𝑇op ((𝑈op 𝑈) −op 𝑈))

Theoremhopncani 29238 Hilbert space operator cancellation law. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
𝑇: ℋ⟶ ℋ    &   𝑈: ℋ⟶ ℋ       ((𝑇 +op 𝑈) −op 𝑈) = 𝑇

Theoremhonpcani 29239 Hilbert space operator cancellation law. (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.)
𝑇: ℋ⟶ ℋ    &   𝑈: ℋ⟶ ℋ       ((𝑇op 𝑈) +op 𝑈) = 𝑇

Theoremhosubeq0i 29240 If the difference between two operators is zero, they are equal. (Contributed by NM, 27-Jul-2006.) (New usage is discouraged.)
𝑇: ℋ⟶ ℋ    &   𝑈: ℋ⟶ ℋ       ((𝑇op 𝑈) = 0hop𝑇 = 𝑈)

Theoremhonpncani 29241 Hilbert space operator cancellation law. (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.)
𝑅: ℋ⟶ ℋ    &   𝑆: ℋ⟶ ℋ    &   𝑇: ℋ⟶ ℋ       ((𝑅op 𝑆) +op (𝑆op 𝑇)) = (𝑅op 𝑇)

Theoremho01i 29242* A condition implying that a Hilbert space operator is identically zero. Lemma 3.2(S8) of [Beran] p. 95. (Contributed by NM, 28-Jan-2006.) (New usage is discouraged.)
𝑇: ℋ⟶ ℋ       (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇𝑥) ·ih 𝑦) = 0 ↔ 𝑇 = 0hop )

Theoremho02i 29243* A condition implying that a Hilbert space operator is identically zero. Lemma 3.2(S10) of [Beran] p. 95. (Contributed by NM, 28-Jan-2006.) (New usage is discouraged.)
𝑇: ℋ⟶ ℋ       (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = 0 ↔ 𝑇 = 0hop )

Theoremhoeq1 29244* A condition implying that two Hilbert space operators are equal. Lemma 3.2(S9) of [Beran] p. 95. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)
((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑆𝑥) ·ih 𝑦) = ((𝑇𝑥) ·ih 𝑦) ↔ 𝑆 = 𝑇))

Theoremhoeq2 29245* A condition implying that two Hilbert space operators are equal. Lemma 3.2(S11) of [Beran] p. 95. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)
((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑆𝑦)) = (𝑥 ·ih (𝑇𝑦)) ↔ 𝑆 = 𝑇))

Theoremadjmo 29246* Every Hilbert space operator has at most one adjoint. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.)
∃*𝑢(𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦))

Theoremadjsym 29247* Symmetry property of an adjoint. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.)
((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑆𝑦)) = ((𝑇𝑥) ·ih 𝑦) ↔ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑆𝑥) ·ih 𝑦)))

Theoremeigrei 29248 A necessary and sufficient condition (that holds when 𝑇 is a Hermitian operator) for an eigenvalue 𝐵 to be real. Generalization of Equation 1.30 of [Hughes] p. 49. (Contributed by NM, 21-Jan-2005.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℂ       (((𝑇𝐴) = (𝐵 · 𝐴) ∧ 𝐴 ≠ 0) → ((𝐴 ·ih (𝑇𝐴)) = ((𝑇𝐴) ·ih 𝐴) ↔ 𝐵 ∈ ℝ))

Theoremeigre 29249 A necessary and sufficient condition (that holds when 𝑇 is a Hermitian operator) for an eigenvalue 𝐵 to be real. Generalization of Equation 1.30 of [Hughes] p. 49. (Contributed by NM, 19-Mar-2006.) (New usage is discouraged.)
(((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ) ∧ ((𝑇𝐴) = (𝐵 · 𝐴) ∧ 𝐴 ≠ 0)) → ((𝐴 ·ih (𝑇𝐴)) = ((𝑇𝐴) ·ih 𝐴) ↔ 𝐵 ∈ ℝ))

Theoremeigposi 29250 A sufficient condition (first conjunct pair, that holds when 𝑇 is a positive operator) for an eigenvalue 𝐵 (second conjunct pair) to be nonnegative. Remark (ii) in [Hughes] p. 137. (Contributed by NM, 2-Jul-2005.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℂ       ((((𝐴 ·ih (𝑇𝐴)) ∈ ℝ ∧ 0 ≤ (𝐴 ·ih (𝑇𝐴))) ∧ ((𝑇𝐴) = (𝐵 · 𝐴) ∧ 𝐴 ≠ 0)) → (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵))

Theoremeigorthi 29251 A necessary and sufficient condition (that holds when 𝑇 is a Hermitian operator) for two eigenvectors 𝐴 and 𝐵 to be orthogonal. Generalization of Equation 1.31 of [Hughes] p. 49. (Contributed by NM, 23-Jan-2005.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ    &   𝐶 ∈ ℂ    &   𝐷 ∈ ℂ       ((((𝑇𝐴) = (𝐶 · 𝐴) ∧ (𝑇𝐵) = (𝐷 · 𝐵)) ∧ 𝐶 ≠ (∗‘𝐷)) → ((𝐴 ·ih (𝑇𝐵)) = ((𝑇𝐴) ·ih 𝐵) ↔ (𝐴 ·ih 𝐵) = 0))

Theoremeigorth 29252 A necessary and sufficient condition (that holds when 𝑇 is a Hermitian operator) for two eigenvectors 𝐴 and 𝐵 to be orthogonal. Generalization of Equation 1.31 of [Hughes] p. 49. (Contributed by NM, 23-Mar-2006.) (New usage is discouraged.)
((((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) ∧ (((𝑇𝐴) = (𝐶 · 𝐴) ∧ (𝑇𝐵) = (𝐷 · 𝐵)) ∧ 𝐶 ≠ (∗‘𝐷))) → ((𝐴 ·ih (𝑇𝐵)) = ((𝑇𝐴) ·ih 𝐵) ↔ (𝐴 ·ih 𝐵) = 0))

19.6.4  Linear, continuous, bounded, Hermitian, unitary operators and norms

Definitiondf-nmop 29253* Define the norm of a Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (New usage is discouraged.)
normop = (𝑡 ∈ ( ℋ ↑𝑚 ℋ) ↦ sup({𝑥 ∣ ∃𝑧 ∈ ℋ ((norm𝑧) ≤ 1 ∧ 𝑥 = (norm‘(𝑡𝑧)))}, ℝ*, < ))

Definitiondf-cnop 29254* Define the set of continuous operators on Hilbert space. For every "epsilon" (𝑦) there is a "delta" (𝑧) such that... (Contributed by NM, 28-Jan-2006.) (New usage is discouraged.)
ContOp = {𝑡 ∈ ( ℋ ↑𝑚 ℋ) ∣ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤 ∈ ℋ ((norm‘(𝑤 𝑥)) < 𝑧 → (norm‘((𝑡𝑤) − (𝑡𝑥))) < 𝑦)}

Definitiondf-lnop 29255* Define the set of linear operators on Hilbert space. (See df-hosum 29144 for definition of operator.) (Contributed by NM, 18-Jan-2006.) (New usage is discouraged.)
LinOp = {𝑡 ∈ ( ℋ ↑𝑚 ℋ) ∣ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑡‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑡𝑦)) + (𝑡𝑧))}

Definitiondf-bdop 29256 Define the set of bounded linear Hilbert space operators. (See df-hosum 29144 for definition of operator.) (Contributed by NM, 18-Jan-2006.) (New usage is discouraged.)
BndLinOp = {𝑡 ∈ LinOp ∣ (normop𝑡) < +∞}

Definitiondf-unop 29257* Define the set of unitary operators on Hilbert space. (Contributed by NM, 18-Jan-2006.) (New usage is discouraged.)
UniOp = {𝑡 ∣ (𝑡: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑡𝑥) ·ih (𝑡𝑦)) = (𝑥 ·ih 𝑦))}

Definitiondf-hmop 29258* Define the set of Hermitian operators on Hilbert space. Some books call these "symmetric operators" and others call them "self-adjoint operators", sometimes with slightly different technical meanings. (Contributed by NM, 18-Jan-2006.) (New usage is discouraged.)
HrmOp = {𝑡 ∈ ( ℋ ↑𝑚 ℋ) ∣ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑡𝑦)) = ((𝑡𝑥) ·ih 𝑦)}

19.6.5  Linear and continuous functionals and norms

Definitiondf-nmfn 29259* Define the norm of a Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)
normfn = (𝑡 ∈ (ℂ ↑𝑚 ℋ) ↦ sup({𝑥 ∣ ∃𝑧 ∈ ℋ ((norm𝑧) ≤ 1 ∧ 𝑥 = (abs‘(𝑡𝑧)))}, ℝ*, < ))

Definitiondf-nlfn 29260 Define the null space of a Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)
null = (𝑡 ∈ (ℂ ↑𝑚 ℋ) ↦ (𝑡 “ {0}))

Definitiondf-cnfn 29261* Define the set of continuous functionals on Hilbert space. For every "epsilon" (𝑦) there is a "delta" (𝑧) such that... (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)
ContFn = {𝑡 ∈ (ℂ ↑𝑚 ℋ) ∣ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤 ∈ ℋ ((norm‘(𝑤 𝑥)) < 𝑧 → (abs‘((𝑡𝑤) − (𝑡𝑥))) < 𝑦)}

Definitiondf-lnfn 29262* Define the set of linear functionals on Hilbert space. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)
LinFn = {𝑡 ∈ (ℂ ↑𝑚 ℋ) ∣ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑡‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑡𝑦)) + (𝑡𝑧))}

Definitiondf-adjh 29263* Define the adjoint of a Hilbert space operator (if it exists). The domain of adj is the set of all adjoint operators. Definition of adjoint in [Kalmbach2] p. 8. Unlike Kalmbach (and most authors), we do not demand that the operator be linear, but instead show (in adjbdln 29497) that the adjoint exists for a bounded linear operator. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.)
adj = {⟨𝑡, 𝑢⟩ ∣ (𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑡𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑢𝑦)))}

19.6.7  Dirac bra-ket notation

Definitiondf-bra 29264* Define the bra of a vector used by Dirac notation. Based on definition of bra in [Prugovecki] p. 186 (p. 180 in 1971 edition). In Dirac bra-ket notation, 𝐴𝐵 is a complex number equal to the inner product (𝐵 ·ih 𝐴). But physicists like to talk about the individual components 𝐴 and 𝐵, called bra and ket respectively. In order for their properties to make sense formally, we define the ket 𝐵 as the vector 𝐵 itself, and the bra 𝐴 as a functional from to . We represent the Dirac notation 𝐴𝐵 by ((bra‘𝐴)‘𝐵); see braval 29358. The reversal of the inner product arguments not only makes the bra-ket behavior consistent with physics literature (see comments under ax-his3 28496) but is also required in order for the associative law kbass2 29531 to work.

Our definition of bra and the associated outer product df-kb 29265 differs from, but is equivalent to, a common approach in the literature that makes use of mappings to a dual space. Our approach eliminates the need to have a parallel development of this dual space and instead keeps everything in Hilbert space.

For an extensive discussion about how our notation maps to the bra-ket notation in physics textbooks, see mmnotes.txt, under the 17-May-2006 entry. (Contributed by NM, 15-May-2006.) (New usage is discouraged.)

bra = (𝑥 ∈ ℋ ↦ (𝑦 ∈ ℋ ↦ (𝑦 ·ih 𝑥)))

Definitiondf-kb 29265* Define a commuted bra and ket juxtaposition used by Dirac notation. In Dirac notation, 𝐴 𝐵 is an operator known as the outer product of 𝐴 and 𝐵, which we represent by (𝐴 ketbra 𝐵). Based on Equation 8.1 of [Prugovecki] p. 376. This definition, combined with definition df-bra 29264, allows any legal juxtaposition of bras and kets to make sense formally and also to obey the associative law when mapped back to Dirac notation. (Contributed by NM, 15-May-2006.) (New usage is discouraged.)
ketbra = (𝑥 ∈ ℋ, 𝑦 ∈ ℋ ↦ (𝑧 ∈ ℋ ↦ ((𝑧 ·ih 𝑦) · 𝑥)))

19.6.8  Positive operators

Definitiondf-leop 29266* Define positive operator ordering. Definition VI.1 of [Retherford] p. 49. Note that ( ℋ × 0) ≤op 𝑇 means that 𝑇 is a positive operator. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
op = {⟨𝑡, 𝑢⟩ ∣ ((𝑢op 𝑡) ∈ HrmOp ∧ ∀𝑥 ∈ ℋ 0 ≤ (((𝑢op 𝑡)‘𝑥) ·ih 𝑥))}

19.6.9  Eigenvectors, eigenvalues, spectrum

Definitiondf-eigvec 29267* Define the eigenvector function. Theorem eleigveccl 29373 shows that eigvec‘𝑇, the set of eigenvectors of Hilbert space operator 𝑇, are Hilbert space vectors. (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.)
eigvec = (𝑡 ∈ ( ℋ ↑𝑚 ℋ) ↦ {𝑥 ∈ ( ℋ ∖ 0) ∣ ∃𝑧 ∈ ℂ (𝑡𝑥) = (𝑧 · 𝑥)})

Definitiondf-eigval 29268* Define the eigenvalue function. The range of eigval‘𝑇 is the set of eigenvalues of Hilbert space operator 𝑇. Theorem eigvalcl 29375 shows that (eigval‘𝑇)‘𝐴, the eigenvalue associated with eigenvector 𝐴, is a complex number. (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.)
eigval = (𝑡 ∈ ( ℋ ↑𝑚 ℋ) ↦ (𝑥 ∈ (eigvec‘𝑡) ↦ (((𝑡𝑥) ·ih 𝑥) / ((norm𝑥)↑2))))

Definitiondf-spec 29269* Define the spectrum of an operator. Definition of spectrum in [Halmos] p. 50. (Contributed by NM, 11-Apr-2006.) (New usage is discouraged.)
Lambda = (𝑡 ∈ ( ℋ ↑𝑚 ℋ) ↦ {𝑥 ∈ ℂ ∣ ¬ (𝑡op (𝑥 ·op ( I ↾ ℋ))): ℋ–1-1→ ℋ})

19.6.10  Theorems about operators and functionals

Theoremnmopval 29270* Value of the norm of a Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
(𝑇: ℋ⟶ ℋ → (normop𝑇) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦)))}, ℝ*, < ))

Theoremelcnop 29271* Property defining a continuous Hilbert space operator. (Contributed by NM, 28-Jan-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
(𝑇 ∈ ContOp ↔ (𝑇: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤 ∈ ℋ ((norm‘(𝑤 𝑥)) < 𝑧 → (norm‘((𝑇𝑤) − (𝑇𝑥))) < 𝑦)))

Theoremellnop 29272* Property defining a linear Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
(𝑇 ∈ LinOp ↔ (𝑇: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑇‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑇𝑦)) + (𝑇𝑧))))

Theoremlnopf 29273 A linear Hilbert space operator is a Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (New usage is discouraged.)
(𝑇 ∈ LinOp → 𝑇: ℋ⟶ ℋ)

Theoremelbdop 29274 Property defining a bounded linear Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
(𝑇 ∈ BndLinOp ↔ (𝑇 ∈ LinOp ∧ (normop𝑇) < +∞))

Theorembdopln 29275 A bounded linear Hilbert space operator is a linear operator. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.)
(𝑇 ∈ BndLinOp → 𝑇 ∈ LinOp)

Theorembdopf 29276 A bounded linear Hilbert space operator is a Hilbert space operator. (Contributed by NM, 2-Feb-2006.) (New usage is discouraged.)
(𝑇 ∈ BndLinOp → 𝑇: ℋ⟶ ℋ)

TheoremnmopsetretALT 29277* The set in the supremum of the operator norm definition df-nmop 29253 is a set of reals. (Contributed by NM, 2-Feb-2006.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝑇: ℋ⟶ ℋ → {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦)))} ⊆ ℝ)

TheoremnmopsetretHIL 29278* The set in the supremum of the operator norm definition df-nmop 29253 is a set of reals. (Contributed by NM, 2-Feb-2006.) (New usage is discouraged.)
(𝑇: ℋ⟶ ℋ → {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦)))} ⊆ ℝ)

Theoremnmopsetn0 29279* The set in the supremum of the operator norm definition df-nmop 29253 is nonempty. (Contributed by NM, 9-Feb-2006.) (New usage is discouraged.)
(norm‘(𝑇‘0)) ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦)))}

Theoremnmopxr 29280 The norm of a Hilbert space operator is an extended real. (Contributed by NM, 9-Feb-2006.) (New usage is discouraged.)
(𝑇: ℋ⟶ ℋ → (normop𝑇) ∈ ℝ*)

Theoremnmoprepnf 29281 The norm of a Hilbert space operator is either real or plus infinity. (Contributed by NM, 5-Feb-2006.) (New usage is discouraged.)
(𝑇: ℋ⟶ ℋ → ((normop𝑇) ∈ ℝ ↔ (normop𝑇) ≠ +∞))

Theoremnmopgtmnf 29282 The norm of a Hilbert space operator is not minus infinity. (Contributed by NM, 2-Feb-2006.) (New usage is discouraged.)
(𝑇: ℋ⟶ ℋ → -∞ < (normop𝑇))

Theoremnmopreltpnf 29283 The norm of a Hilbert space operator is real iff it is less than infinity. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
(𝑇: ℋ⟶ ℋ → ((normop𝑇) ∈ ℝ ↔ (normop𝑇) < +∞))

Theoremnmopre 29284 The norm of a bounded operator is a real number. (Contributed by NM, 29-Jan-2006.) (New usage is discouraged.)
(𝑇 ∈ BndLinOp → (normop𝑇) ∈ ℝ)

Theoremelbdop2 29285 Property defining a bounded linear Hilbert space operator. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
(𝑇 ∈ BndLinOp ↔ (𝑇 ∈ LinOp ∧ (normop𝑇) ∈ ℝ))

Theoremelunop 29286* Property defining a unitary Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (New usage is discouraged.)
(𝑇 ∈ UniOp ↔ (𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇𝑥) ·ih (𝑇𝑦)) = (𝑥 ·ih 𝑦)))

Theoremelhmop 29287* Property defining a Hermitian Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
(𝑇 ∈ HrmOp ↔ (𝑇: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑇𝑥) ·ih 𝑦)))

Theoremhmopf 29288 A Hermitian operator is a Hilbert space operator (mapping). (Contributed by NM, 19-Mar-2006.) (New usage is discouraged.)
(𝑇 ∈ HrmOp → 𝑇: ℋ⟶ ℋ)

Theoremhmopex 29289 The class of Hermitian operators is a set. (Contributed by NM, 17-Aug-2006.) (New usage is discouraged.)
HrmOp ∈ V

Theoremnmfnval 29290* Value of the norm of a Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
(𝑇: ℋ⟶ℂ → (normfn𝑇) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇𝑦)))}, ℝ*, < ))

Theoremnmfnsetre 29291* The set in the supremum of the functional norm definition df-nmfn 29259 is a set of reals. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
(𝑇: ℋ⟶ℂ → {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇𝑦)))} ⊆ ℝ)

Theoremnmfnsetn0 29292* The set in the supremum of the functional norm definition df-nmfn 29259 is nonempty. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
(abs‘(𝑇‘0)) ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇𝑦)))}

Theoremnmfnxr 29293 The norm of any Hilbert space functional is an extended real. (Contributed by NM, 9-Feb-2006.) (New usage is discouraged.)
(𝑇: ℋ⟶ℂ → (normfn𝑇) ∈ ℝ*)

Theoremnmfnrepnf 29294 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 → ((normfn𝑇) ∈ ℝ ↔ (normfn𝑇) ≠ +∞))

Theoremnlfnval 29295 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}))

Theoremelcnfn 29296* 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‘(𝑤 𝑥)) < 𝑧 → (abs‘((𝑇𝑤) − (𝑇𝑥))) < 𝑦)))

Theoremellnfn 29297* 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 · 𝑦) + 𝑧)) = ((𝑥 · (𝑇𝑦)) + (𝑇𝑧))))

Theoremlnfnf 29298 A linear Hilbert space functional is a functional. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.)
(𝑇 ∈ LinFn → 𝑇: ℋ⟶ℂ)

Theoremdfadj2 29299* Alternate definition of the adjoint of a Hilbert space operator. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.)
adj = {⟨𝑡, 𝑢⟩ ∣ (𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑡𝑦)) = ((𝑢𝑥) ·ih 𝑦))}

Theoremfunadj 29300 Functionality of the adjoint function. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)