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Definition df-hmo 28575
 Description: Define the set of Hermitian (self-adjoint) operators on a normed complex vector space (normally a Hilbert space). Although we define it for any normed vector space for convenience, the definition is meaningful only for inner product spaces. (Contributed by NM, 26-Jan-2008.) (New usage is discouraged.)
Assertion
Ref Expression
df-hmo HmOp = (𝑢 ∈ NrmCVec ↦ {𝑡 ∈ dom (𝑢adj𝑢) ∣ ((𝑢adj𝑢)‘𝑡) = 𝑡})
Distinct variable group:   𝑢,𝑡

Detailed syntax breakdown of Definition df-hmo
StepHypRef Expression
1 chmo 28573 . 2 class HmOp
2 vu . . 3 setvar 𝑢
3 cnv 28408 . . 3 class NrmCVec
4 vt . . . . . . 7 setvar 𝑡
54cv 1537 . . . . . 6 class 𝑡
62cv 1537 . . . . . . 7 class 𝑢
7 caj 28572 . . . . . . 7 class adj
86, 6, 7co 7142 . . . . . 6 class (𝑢adj𝑢)
95, 8cfv 6329 . . . . 5 class ((𝑢adj𝑢)‘𝑡)
109, 5wceq 1538 . . . 4 wff ((𝑢adj𝑢)‘𝑡) = 𝑡
118cdm 5522 . . . 4 class dom (𝑢adj𝑢)
1210, 4, 11crab 3110 . . 3 class {𝑡 ∈ dom (𝑢adj𝑢) ∣ ((𝑢adj𝑢)‘𝑡) = 𝑡}
132, 3, 12cmpt 5113 . 2 class (𝑢 ∈ NrmCVec ↦ {𝑡 ∈ dom (𝑢adj𝑢) ∣ ((𝑢adj𝑢)‘𝑡) = 𝑡})
141, 13wceq 1538 1 wff HmOp = (𝑢 ∈ NrmCVec ↦ {𝑡 ∈ dom (𝑢adj𝑢) ∣ ((𝑢adj𝑢)‘𝑡) = 𝑡})
 Colors of variables: wff setvar class This definition is referenced by:  hmoval  28634
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