definition 12.2 Hilbert space

open in the book · parts/02-mathematical-methods/10-hilbert-spaces.tex:67 · p. 413

ground object -- no derivation owed

Rests on

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 4.3 Algebraic structure ¶
      - depends_on definition 4.2 Binary operation ¶
  - depends_on definition 4.32 Field ¶
    - depends_on definition 4.11 Inverse element ¶
      - depends_on definition 4.9 Neutral element ¶
    - depends_on definition 4.20 Monoid ¶
      - depends_on definition 4.9 Neutral element ¶ ↺
      - depends_on definition 4.18 Semigroup ¶
    - depends_on definition 4.30 Ring ¶
      - depends_on definition 4.8 Commutativity; abelian structure ¶ ↺
      - depends_on definition 4.28 Distributivity ¶
      - depends_on definition 4.21 Group ¶
      - depends_on definition 4.18 Semigroup ¶ ↺
  - depends_on definition 4.31 Module ¶
    - depends_on definition 4.29 Action ¶
      - depends_on definition 4.9 Neutral element ¶ ↺
    - 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 3.23 Predicate ¶
    - 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 definition A.278 Countably Hilbert nuclear space ¶
- depends_on theorem A.281 Nuclear spaces embed by Hilbert–Schmidt maps; quoted ¶
  - depends_on proposition A.282 The fibre maps are continuous on $\Phi$ ¶
depends_on definition 12.32 Separable Hilbert space ¶
- depends_on lemma A.250 Decomposition into cyclic subspaces ¶
  - depends_on proposition A.280 Direct-integral form of the spectral theorem ¶
    - depends_on proposition A.282 The fibre maps are continuous on $\Phi$ ¶ ↺
depends_on definition 12.94 Tensor product of Hilbert spaces ¶
- depends_on definition 12.99 Product and entangled vectors ¶
  - depends_on example 12.100 Entangled vectors exist ¶
- depends_on example 12.98 Two particles in three-dimensional space ¶
- depends_on proposition 12.95 The tensor inner product is well defined and positive definite ¶
  - depends_on example 12.100 Entangled vectors exist ¶ ↺
  - depends_on proposition 12.96 Operators on a tensor product ¶
depends_on definition 10.85 Sobolev space ¶
- depends_on definition A.450 The space $W^{1,r}(a,b)$ ¶
  - depends_on definition A.462 The weighted space and the energy space ¶
    - depends_on lemma A.463 Poincaré inequality; $B_{K}$ is an inner product ¶
      - depends_on lemma A.464 $H_{E}$ is a Hilbert space ¶
      - depends_on lemma A.466 The homogeneous Dirichlet problem is trivial ¶
      - depends_on theorem A.469 The Green operator inverts $L_{K}$ ¶
  - depends_on lemma A.453 Uniform bound and uniform Hölder continuity ¶
    - depends_on theorem A.455 Compact embedding of $W^{1,r}(a,b)$ into the continuous functions ¶
      - depends_on theorem A.456 Convexity in $q$ implies weak lower semicontinuity ¶
  - depends_on lemma A.452 Young and Hölder ¶
    - depends_on lemma A.489 Courant–Lebesgue ¶
      - depends_on theorem A.490 Equicontinuity of the normalised class ¶
    - depends_on lemma A.463 Poincaré inequality; $B_{K}$ is an inner product ¶ ↺
    - depends_on lemma A.453 Uniform bound and uniform Hölder continuity ¶ ↺
- depends_on definition 10.87 Weak form of an elliptic problem ¶
  - depends_on corollary 10.89 Existence for the elliptic problem ¶
  - depends_on proposition 10.91 Dirichlet principle ¶
  - depends_on theorem 10.88 Lax–Milgram ¶
    - depends_on corollary 10.89 Existence for the elliptic problem ¶ ↺
    - depends_on proposition 10.91 Dirichlet principle ¶ ↺
depends_on example 12.9 The sequence space $\ell^{2}$ ¶
- depends_on definition 12.84 External direct sum ¶
  - depends_on proposition 12.85 The direct sum is a Hilbert space ¶
    - depends_on proposition 12.87 Expansion in an orthogonal decomposition ¶
      - depends_on lemma A.250 Decomposition into cyclic subspaces ¶ ↺
- depends_on proposition 12.10 $\ell^{2}$ is complete ¶
  - depends_on proposition 12.85 The direct sum is a Hilbert space ¶ ↺
  - depends_on theorem 12.33 Every separable Hilbert space is $\ell^{2}$ ¶
depends_on example 12.11 The function space $L^{2}$ ¶
- 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 ¶ ↺
depends_on lemma A.254 Riemann integral of a continuous curve ¶
- depends_on lemma A.255 Smoothed vectors lie in the domain ¶
  - depends_on proposition A.256 Density ¶
    - depends_on proposition A.259 The generator is self-adjoint ¶
      - depends_on lemma A.260 Cayley transform of a self-adjoint operator ¶
      - depends_on proposition A.263 The two constructions are inverse ¶
- depends_on lemma A.257 Integrated form of the equation of motion ¶
  - depends_on proposition A.258 The generator is closed ¶
    - depends_on proposition A.259 The generator is self-adjoint ¶ ↺
- depends_on lemma A.583 Absolutely convergent operator-valued integrals ¶
  - depends_on definition A.585 The Gaussian average of the Weyl operators ¶
    - depends_on theorem A.587 The Gaussian average is a non-zero projector that absorbs the Weyl operators ¶
      - depends_on lemma A.590 The Gram matrix is universal ¶
      - depends_on proposition A.589 Cyclic subspaces and the rank of the average ¶
      - depends_on proposition A.592 The vacuum of the Schrödinger system ¶ ↺
depends_on proposition 12.4 Cauchy–Schwarz and continuity of the inner product ¶
- depends_on corollary 12.5 Continuity of the norm and of orthogonality ¶
  - depends_on proposition 12.17 The complement is always a closed subspace ¶
    - depends_on corollary 12.19 Double complement; the density criterion ¶
      - depends_on definition 12.79 Deficiency subspaces and indices ¶
      - depends_on definition 12.69 Operator with a domain ¶
      - depends_on definition 12.103 Gelfand triple ¶
      - depends_on definition 12.50 Point, continuous and residual spectrum ¶
      - depends_on definition 12.71 Adjoint of a densely defined operator ¶
      - depends_on theorem 12.30 Completeness, expansion, Parseval ¶
      - depends_on theorem 12.55 The spectrum of a self-adjoint operator is real ¶
    - depends_on theorem 12.18 Projection theorem ¶
      - depends_on corollary 12.19 Double complement; the density criterion ¶ ↺
      - depends_on definition A.248 Cyclic vector and cyclic subspace ¶
      - depends_on definition 12.20 Orthogonal projection operator ¶
      - depends_on lemma A.233 Construction of the system ¶
      - depends_on lemma A.231 Restriction to an invariant closed subspace ¶
      - depends_on lemma A.267 Isometry of $A\pm\ii\mu$, and closed range ¶
      - depends_on proposition 12.27 Best approximation and Bessel's inequality ¶
      - depends_on proposition 12.21 Characterization of orthogonal projections ¶
      - depends_on theorem 12.46 Riesz representation ¶
- depends_on lemma A.606 $F$ is entire, and is the overlap in disguise ¶
  - depends_on lemma A.607 A zero-free entire function is an exponential ¶
    - depends_on proposition A.609 The exponent is a quadratic polynomial ¶
      - depends_on proposition A.611 The constraint on $\Gamma$ ¶
- depends_on lemma A.254 Riemann integral of a continuous curve ¶ ↺
- depends_on lemma A.583 Absolutely convergent operator-valued integrals ¶ ↺
- depends_on proposition 12.102 Neither plane waves nor deltas are in $L^{2}$ ¶ ↺
- depends_on theorem 25.53 Properties of the Wigner function ¶
  - depends_on lemma A.603 Overlaps and the resolution of unity ¶
    - depends_on proposition A.611 The constraint on $\Gamma$ ¶ ↺
    - depends_on proposition A.604 Non-negativity forbids a vanishing overlap ¶
      - depends_on proposition A.609 The exponent is a quadratic polynomial ¶ ↺
depends_on proposition 12.28 Convergence criterion for orthogonal series ¶
- depends_on theorem 12.30 Completeness, expansion, Parseval ¶ ↺
depends_on proposition 12.51 The three cases are exclusive and exhaustive ¶
depends_on theorem 16.70 The direct method ¶
- depends_on lemma 16.72 Convexity implies weak lower semicontinuity ¶
  - depends_on theorem A.448 Tonelli ¶
  - depends_on theorem A.456 Convexity in $q$ implies weak lower semicontinuity ¶ ↺
  - depends_on theorem 16.73 Tonelli ¶
- depends_on theorem A.477 Douglas; Radó ¶
- depends_on theorem A.448 Tonelli ¶ ↺
- depends_on theorem 16.81 Douglas; Radó ¶
- depends_on theorem 16.73 Tonelli ¶ ↺
depends_on theorem 12.14 Closest point in a closed convex set ¶
- depends_on theorem 12.18 Projection theorem ¶ ↺
… 1 more

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`	→	Inner product	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:74
`depends_on`	→	Convergence; Cauchy sequence; completeness	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:74
`depends_on`	→	eq:lin-norm-assoc	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:74
`depends_on`	←	Countably Hilbert nuclear space	declared	appendices/A-long-proofs.tex:13903
`depends_on`	←	Separable Hilbert space	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:817
`depends_on`	←	Tensor product of Hilbert spaces	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2596
`depends_on`	←	Sobolev space	declared	parts/02-mathematical-methods/08-pdes.tex:2295
`depends_on`	←	The sequence space $\ell^{2}$	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:245
`depends_on`	←	The function space $L^{2}$	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:286
`depends_on`	←	Riemann integral of a continuous curve	declared	appendices/A-long-proofs.tex:12667
`depends_on`	←	Cauchy–Schwarz and continuity of the inner product	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:115
`depends_on`	←	Convergence criterion for orthogonal series	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:708
`depends_on`	←	The three cases are exclusive and exhaustive	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1332
`depends_on`	←	The direct method	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:2184
`depends_on`	←	Closest point in a closed convex set	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:348
`depends_on`	←	Lax–Milgram	declared	parts/02-mathematical-methods/08-pdes.tex:2350