example 12.11 The function space $L^{2}$

open in the book · parts/02-mathematical-methods/10-hilbert-spaces.tex:272 · p. 416

example -- no derivation owed by its kind

Rests on

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 ¶
depends_on definition 5.18 Inner product ¶ ↺

Supports

depends_on example 12.81 Momentum on a finite interval: a circle of self-adjoint momenta ¶
- depends_on corollary A.274 Momentum on ${[}0,L{]}$: the circle of extensions ¶
- depends_on example 12.82 Momentum on the half-line: no self-adjoint extension ¶
  - depends_on corollary A.275 Momentum on $[0,\infty)$: no extension ¶
depends_on example 12.56 Multiplication by the coordinate: spectrum without eigenvectors ¶
depends_on example 12.25 Orthogonal polynomials ¶
depends_on example 12.110 The Schrödinger system ¶
- depends_on proposition A.592 The vacuum of the Schrödinger system ¶
- depends_on theorem A.579 Stone–von Neumann ¶
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 ¶ ↺
depends_on example 12.98 Two particles in three-dimensional space ¶
depends_on proposition 12.102 Neither plane waves nor deltas are in $L^{2}$ ¶
- depends_on example 12.104 The Schwartz triple ¶ ↺
depends_on theorem 12.12 Riesz–Fischer ¶
- depends_on lemma A.464 $H_{E}$ is a Hilbert space ¶

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`	→	Hilbert space	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:286
`depends_on`	→	Inner product	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:286
`depends_on`	←	Momentum on a finite interval: a circle of self-adjoint momenta	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2225
`depends_on`	←	Multiplication by the coordinate: spectrum without eigenvectors	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1504
`depends_on`	←	Orthogonal polynomials	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:637
`depends_on`	←	The Schrödinger system	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:3038
`depends_on`	←	The Schwartz triple	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2844
`depends_on`	←	Two particles in three-dimensional space	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2720
`depends_on`	←	Neither plane waves nor deltas are in $L^{2}$	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2788
`depends_on`	←	Riesz–Fischer	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:291