theorem A.281 Nuclear spaces embed by Hilbert–Schmidt maps; quoted

open in the book · appendices/A-long-proofs.tex:14027 · p. 2929

Rests on

depends_on definition A.278 Countably Hilbert nuclear space ¶
- 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 5.46 Functional ¶
    - 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-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 corollary 12.5 Continuity of the norm and of orthogonality ¶
      - depends_on definition 12.16 Orthogonal complement ¶
      - proves proof ch:10-hilbert-spaces@proof-8 ¶
    - 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 4.32 Field ¶
      - depends_on definition 4.31 Module ¶
  - depends_on definition 6.27 Convergence; Cauchy sequence; completeness ¶
    - depends_on definition 6.24 Metric ¶
      - depends_on definition 3.43 Map ¶
      - depends_on definition 3.38 Ordered pair and Cartesian product ¶
  - depends_on equation 5.44 eq:lin-norm-assoc ¶

Supports

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`	→	Countably Hilbert nuclear space	declared	appendices/A-long-proofs.tex:14038
`depends_on`	←	The fibre maps are continuous on $\Phi$	declared	appendices/A-long-proofs.tex:14062