example 12.104 The Schwartz triple

open in the book · parts/02-mathematical-methods/10-hilbert-spaces.tex:2825 · p. 443

example -- no derivation owed by its kind

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 example 12.11 The function space $L^{2}$ ¶
- 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 ¶
- depends_on definition 5.18 Inner product ¶ ↺
depends_on proposition 12.102 Neither plane waves nor deltas are in $L^{2}$ ¶
- depends_on example 12.11 The function space $L^{2}$ ¶ ↺
- 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 5.5 Linear combination ¶
    - 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 ch:10-hilbert-spaces@proof-48 ¶

Supports

depends_on example A.283 Momentum on the line ¶
depends_on proposition 12.106 The plane wave is a generalized momentum eigenvector ¶
- depends_on example A.283 Momentum on the line ¶ ↺

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
`cites`	→	Théorie des distributions	derived	parts/02-mathematical-methods/10-hilbert-spaces.tex:2832
`depends_on`	→	Gelfand triple	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2844
`depends_on`	→	The function space $L^{2}$	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2844
`depends_on`	→	Neither plane waves nor deltas are in $L^{2}$	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2844
`depends_on`	←	Momentum on the line	declared	appendices/A-long-proofs.tex:14226
`depends_on`	←	The plane wave is a generalized momentum eigenvector	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2867