theorem A.242 Stone–Weierstrass; quoted

open in the book · appendices/A-long-proofs.tex:12091 · p. 2909

Rests on

depends_on definition 6.9 Compact set ¶
- depends_on definition 6.5 Open cover ¶
  - 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 3.30 Subset ¶
      - depends_on equation 3.51 eq:set-indexed ¶
  - depends_on equation 3.51 eq:set-indexed ¶ ↺

Supports

depends_on proposition A.243 Continuous functional calculus ¶
- depends_on definition A.248 Cyclic vector and cyclic subspace ¶
  - depends_on lemma A.249 The cyclic case ¶
  - depends_on lemma A.250 Decomposition into cyclic subspaces ¶
    - 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.246 Bounded Borel functional calculus ¶
  - 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.261 Spectral theorem for a unitary operator ¶
    - depends_on proposition A.262 Spectral theorem for an unbounded self-adjoint operator ¶ ↺
- depends_on proposition A.245 The measures $\mu_{x,y}$ ¶
  - depends_on lemma A.249 The cyclic case ¶ ↺
  - depends_on proposition A.246 Bounded Borel functional calculus ¶ ↺

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`	→	Compact set	declared	appendices/A-long-proofs.tex:12100
`depends_on`	←	Continuous functional calculus	declared	appendices/A-long-proofs.tex:12117