definition A.278 Countably Hilbert nuclear space

open in the book · appendices/A-long-proofs.tex:13891 · p. 2928

ground object -- no derivation owed

Rests on

depends_on definition 12.103 Gelfand triple ¶
- depends_on corollary 12.47 $\mathcal{H}$ is its own dual, antilinearly ¶
  - depends_on definition 12.45 Continuous linear functional; the dual ¶
    - depends_on definition 12.35 Bounded operator; operator norm ¶
      - depends_on definition 5.37 Linear transformation ¶
      - depends_on definition 5.19 Norm ¶
    - depends_on definition 5.46 Functional ¶
      - depends_on definition 5.37 Linear transformation ¶ ↺
  - depends_on theorem 12.46 Riesz representation ¶
    - depends_on definition 12.45 Continuous linear functional; the dual ¶ ↺
    - depends_on theorem 12.18 Projection theorem ¶
      - depends_on proposition 12.17 The complement is always a closed subspace ¶
      - depends_on theorem 12.14 Closest point in a closed convex set ¶
      - proves proof ch:10-hilbert-spaces@proof-9 ¶
    - proves proof ch:10-hilbert-spaces@proof-24 ¶
  - proves proof ch:10-hilbert-spaces@proof-25 ¶
- depends_on corollary 12.19 Double complement; the density criterion ¶
  - depends_on proposition 12.17 The complement is always a closed subspace ¶ ↺
  - depends_on theorem 12.18 Projection theorem ¶ ↺
  - proves proof ch:10-hilbert-spaces@proof-10 ¶
- depends_on definition 12.45 Continuous linear functional; the dual ¶ ↺
depends_on definition 12.2 Hilbert space ¶
- depends_on definition 5.18 Inner product ¶
  - depends_on definition 4.33 Vector space ¶
    - depends_on definition 4.8 Commutativity; abelian structure ¶
      - depends_on definition 4.4 Internal binary operation; magma ¶
    - depends_on definition 4.32 Field ¶
      - depends_on definition 4.11 Inverse element ¶
      - depends_on definition 4.20 Monoid ¶
      - depends_on definition 4.30 Ring ¶
    - depends_on definition 4.31 Module ¶
      - depends_on definition 4.29 Action ¶
      - depends_on definition 4.28 Distributivity ¶
      - depends_on definition 4.21 Group ¶
      - depends_on definition 4.30 Ring ¶ ↺
- depends_on definition 6.27 Convergence; Cauchy sequence; completeness ¶
  - depends_on definition 6.24 Metric ¶
    - depends_on definition 3.43 Map ¶
      - depends_on definition 3.24 Quantifiers ¶
      - depends_on definition 3.28 Set ¶
    - depends_on definition 3.38 Ordered pair and Cartesian product ¶
- depends_on equation 5.44 eq:lin-norm-assoc ¶

Supports

depends_on theorem A.281 Nuclear spaces embed by Hilbert–Schmidt maps; quoted ¶
- depends_on proposition A.282 The fibre maps are continuous on $\Phi$ ¶

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`	→	Gelfand triple	declared	appendices/A-long-proofs.tex:13903
`depends_on`	→	Hilbert space	declared	appendices/A-long-proofs.tex:13903
`depends_on`	←	Nuclear spaces embed by Hilbert–Schmidt maps; quoted	declared	appendices/A-long-proofs.tex:14038