proposition A.256 Density

open in the book · appendices/A-long-proofs.tex:12746 · p. 2916

Rests on

depends_on definition 12.64 Strongly continuous one-parameter unitary group ¶
- depends_on definition 12.41 The operator classes ¶
  - depends_on definition 6.9 Compact set ¶
    - depends_on definition 6.5 Open cover ¶
      - depends_on definition 6.2 Open set ¶
      - depends_on equation 3.51 eq:set-indexed ¶
  - depends_on proposition 12.21 Characterization of orthogonal projections ¶
    - depends_on definition 12.20 Orthogonal projection operator ¶
      - depends_on theorem 12.18 Projection theorem ¶
    - depends_on theorem 12.18 Projection theorem ¶ ↺
    - proves proof ch:10-hilbert-spaces@proof-11 ¶
  - depends_on theorem 12.38 Existence and uniqueness of the adjoint ¶
    - depends_on definition 5.41 Adjoint ¶
      - depends_on definition 5.18 Inner product ¶
      - depends_on definition 5.37 Linear transformation ¶
    - depends_on proposition 12.37 $\mathcal{B}(\mathcal{H})$ is a Banach algebra ¶
      - depends_on definition 12.35 Bounded operator; operator norm ¶
      - depends_on proposition 12.8 Absolutely convergent series test ¶
      - proves proof ch:10-hilbert-spaces@proof-19 ¶
    - depends_on theorem 12.46 Riesz representation ¶
      - depends_on definition 12.45 Continuous linear functional; the dual ¶
      - depends_on theorem 12.18 Projection theorem ¶ ↺
      - proves proof ch:10-hilbert-spaces@proof-24 ¶
    - proves proof ch:10-hilbert-spaces@proof-20 ¶
- depends_on definition 6.6 Continuous map ¶
  - depends_on definition 3.53 Preimage ¶
  - depends_on definition 6.2 Open set ¶ ↺
  - depends_on definition 6.1 Topological space ¶
    - depends_on definition 3.31 Empty set ¶
    - depends_on definition 3.35 Union, intersection, difference ¶
      - depends_on definition 3.8 Conjunction ¶
      - depends_on definition 3.9 Disjunction ¶
      - depends_on definition 3.7 Negation ¶
    - depends_on definition 3.30 Subset ¶
    - depends_on equation 3.51 eq:set-indexed ¶ ↺
depends_on lemma A.255 Smoothed vectors lie in the domain ¶
- depends_on definition 12.64 Strongly continuous one-parameter unitary group ¶ ↺
- depends_on equation A.470 eq:app-stone-domain ¶
- depends_on lemma A.254 Riemann integral of a continuous curve ¶
  - depends_on definition 12.2 Hilbert space ¶
    - depends_on definition 5.18 Inner product ¶ ↺
    - depends_on definition 6.27 Convergence; Cauchy sequence; completeness ¶
      - depends_on definition 6.24 Metric ¶
    - depends_on equation 5.44 eq:lin-norm-assoc ¶
  - depends_on proposition 12.4 Cauchy–Schwarz and continuity of the inner product ¶
    - depends_on definition 12.2 Hilbert space ¶ ↺
    - depends_on proposition 5.20 Cauchy–Schwarz inequality ¶
      - depends_on definition 5.14 Linear independence ¶
      - depends_on definition 5.18 Inner product ¶ ↺
      - proves proof ch:03-linear-algebra-representations@proof-4 ¶
    - proves proof ch:10-hilbert-spaces@proof-1 ¶
  - proves proof app:A-long-proofs@proof-157 ¶
- proves proof app:A-long-proofs@proof-158 ¶
proves proof app:A-long-proofs@proof-159 ¶

Supports

depends_on proposition A.259 The generator is self-adjoint ¶
- depends_on lemma A.260 Cayley transform of a self-adjoint operator ¶
  - depends_on proposition A.262 Spectral theorem for an unbounded self-adjoint operator ¶
    - depends_on proposition A.280 Direct-integral form of the spectral theorem ¶
      - depends_on proposition A.282 The fibre maps are continuous on $\Phi$ ¶
- depends_on proposition A.263 The two constructions are inverse ¶

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Strongly continuous one-parameter unitary group	declared	appendices/A-long-proofs.tex:12748
`depends_on`	→	Smoothed vectors lie in the domain	declared	appendices/A-long-proofs.tex:12748
`depends_on`	←	The generator is self-adjoint	declared	appendices/A-long-proofs.tex:12845
`proves`	←	app:A-long-proofs@proof-159	declared	appendices/A-long-proofs.tex:12751