proposition 12.102 Neither plane waves nor deltas are in $L^{2}$

open in the book · parts/02-mathematical-methods/10-hilbert-spaces.tex:2782 · p. 443

Rests on

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 12.4 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 4.33 Vector space ¶ ↺
  - 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 12.104 The Schwartz triple ¶
- 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
`depends_on`	→	The function space $L^{2}$	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2788
`depends_on`	→	Cauchy–Schwarz and continuity of the inner product	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2788
`depends_on`	←	The Schwartz triple	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2844
`proves`	←	ch:10-hilbert-spaces@proof-48	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2791