definition 12.16 Orthogonal complement

open in the book · parts/02-mathematical-methods/10-hilbert-spaces.tex:392 · p. 417

ground object -- no derivation owed

Rests on

depends_on definition 5.24 Orthogonal vectors ¶
- 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 equation 5.52 eq:lin-orthogonal-complement ¶

Supports

depends_on definition 12.29 Orthonormal basis ¶
- depends_on definition 12.86 Internal orthogonal decomposition ¶
  - depends_on proposition 12.87 Expansion in an orthogonal decomposition ¶
    - depends_on lemma A.250 Decomposition into cyclic subspaces ¶
      - depends_on proposition A.280 Direct-integral form of the spectral theorem ¶
depends_on definition 12.86 Internal orthogonal decomposition ¶ ↺
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 A.269 Cayley transform ¶
      - depends_on proposition A.270 Properties of the transform ¶
    - 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 12.81 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 lemma A.267 Isometry of $A\pm\ii\mu$, and closed range ¶
      - depends_on definition A.269 Cayley transform ¶ ↺
      - depends_on lemma A.268 The indices do not depend on $\mu$ ¶
      - depends_on proposition A.270 Properties of the transform ¶ ↺
      - depends_on proposition A.273 Self-adjoint means unitary ¶
    - depends_on lemma A.268 The indices do not depend on $\mu$ ¶ ↺
    - depends_on theorem A.266 von Neumann ¶
      - depends_on corollary A.275 Momentum on $[0,\infty)$: no extension ¶ ↺
      - depends_on corollary A.274 Momentum on ${[}0,L{]}$: the circle of extensions ¶ ↺
    - depends_on theorem 12.80 von Neumann's criterion ¶
      - depends_on example 12.82 Momentum on the half-line: no self-adjoint extension ¶ ↺
      - depends_on example 12.81 Momentum on a finite interval: a circle of self-adjoint momenta ¶ ↺
  - depends_on definition 12.69 Operator with a domain ¶
    - depends_on definition 12.70 Graph; closed and closable operators ¶
      - depends_on proposition 12.73 The adjoint is always closed ¶
    - depends_on definition 12.72 Symmetric; self-adjoint ¶
      - depends_on definition 12.78 Essential self-adjointness ¶
      - depends_on lemma A.260 Cayley transform of a self-adjoint operator ¶
      - depends_on lemma A.267 Isometry of $A\pm\ii\mu$, and closed range ¶ ↺
      - depends_on lemma A.271 Injectivity of $\identity-V$ for any isometric extension ¶
      - depends_on lemma A.272 The operator attached to an isometry ¶
      - depends_on proposition A.273 Self-adjoint means unitary ¶ ↺
      - depends_on theorem A.266 von Neumann ¶ ↺
      - depends_on theorem 12.74 Hellinger–Toeplitz ¶
      - depends_on theorem 12.80 von Neumann's criterion ¶ ↺
    - depends_on definition 12.71 Adjoint of a densely defined operator ¶
      - depends_on definition 12.79 Deficiency subspaces and indices ¶ ↺
      - depends_on definition 12.72 Symmetric; self-adjoint ¶ ↺
      - depends_on proposition 12.73 The adjoint is always closed ¶ ↺
  - depends_on definition 12.103 Gelfand triple ¶
    - depends_on definition A.278 Countably Hilbert nuclear space ¶
      - depends_on theorem A.281 Nuclear spaces embed by Hilbert–Schmidt maps; quoted ¶
    - depends_on definition 12.105 Generalized eigenvector ¶
      - depends_on proposition 12.106 The plane wave is a generalized momentum eigenvector ¶
      - depends_on theorem A.279 Gelfand–Maurin ¶
      - depends_on theorem 12.107 Nuclear spectral theorem ¶
    - 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 theorem A.279 Gelfand–Maurin ¶ ↺
    - depends_on theorem 12.107 Nuclear spectral theorem ¶ ↺
  - depends_on definition 12.50 Point, continuous and residual spectrum ¶
    - depends_on definition 12.105 Generalized eigenvector ¶ ↺
    - depends_on example 12.56 Multiplication by the coordinate: spectrum without eigenvectors ¶
    - depends_on proposition 12.51 The three cases are exclusive and exhaustive ¶
  - depends_on definition 12.71 Adjoint of a densely defined operator ¶ ↺
  - depends_on theorem 12.30 Completeness, expansion, Parseval ¶
    - depends_on proposition 17.26 The lattice harmonics are an orthonormal basis ¶
    - depends_on proposition 12.87 Expansion in an orthogonal decomposition ¶ ↺
    - 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 theorem A.229 Hilbert–Schmidt ¶
    - depends_on theorem 12.44 Hilbert–Schmidt: compact self-adjoint operators ¶
      - depends_on theorem A.461 Completeness in the weighted and in the energy norm ¶
      - depends_on theorem A.471 Spectral decomposition and completeness in $L^{2}_{r}$ ¶
    - depends_on theorem 12.33 Every separable Hilbert space is $\ell^{2}$ ¶
  - depends_on theorem 12.55 The spectrum of a self-adjoint operator is real ¶
    - depends_on example 12.56 Multiplication by the coordinate: spectrum without eigenvectors ¶ ↺
    - depends_on theorem A.238 Spectral theorem, both forms ¶
      - depends_on proposition A.280 Direct-integral form of the spectral theorem ¶ ↺
      - depends_on proposition A.261 Spectral theorem for a unitary operator ¶
      - depends_on theorem A.279 Gelfand–Maurin ¶ ↺
      - depends_on theorem A.253 Stone ¶
    - depends_on theorem 12.59 Spectral theorem for a bounded self-adjoint operator ¶
      - depends_on definition 12.60 Functional calculus ¶
      - depends_on theorem 12.107 Nuclear spectral theorem ¶ ↺
      - depends_on theorem 12.91 Schur's lemma, commutant form ¶
      - depends_on theorem 12.66 Stone ¶
- 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 lemma A.249 The cyclic case ¶
    - depends_on lemma A.250 Decomposition into cyclic subspaces ¶ ↺
  - depends_on definition 12.20 Orthogonal projection operator ¶
    - depends_on definition 12.88 Reducing subspace ¶
      - depends_on definition 12.90 Self-adjoint family; commutant; irreducibility ¶
      - depends_on proposition 12.89 Reduction is commutation ¶
    - depends_on proposition 12.21 Characterization of orthogonal projections ¶
      - depends_on definition 12.41 The operator classes ¶
      - depends_on definition 12.58 Projection-valued measure ¶
      - depends_on proposition 12.89 Reduction is commutation ¶ ↺
  - depends_on lemma A.233 Construction of the system ¶
    - depends_on lemma A.234 The eigenvalues tend to zero, with finite multiplicity ¶
  - depends_on lemma A.231 Restriction to an invariant closed subspace ¶
    - depends_on lemma A.233 Construction of the system ¶ ↺
  - 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 theorem 12.30 Completeness, expansion, Parseval ¶ ↺
  - depends_on proposition 12.21 Characterization of orthogonal projections ¶ ↺
  - depends_on theorem 12.46 Riesz representation ¶
    - depends_on corollary 12.47 $\mathcal{H}$ is its own dual, antilinearly ¶
      - depends_on definition 12.103 Gelfand triple ¶ ↺
    - depends_on lemma A.247 Integration against a projection-valued measure ¶
    - depends_on proposition A.246 Bounded Borel functional calculus ¶
      - depends_on proposition A.262 Spectral theorem for an unbounded self-adjoint operator ¶
      - depends_on proposition A.261 Spectral theorem for a unitary operator ¶ ↺
    - depends_on proposition A.245 The measures $\mu_{x,y}$ ¶
      - depends_on lemma A.249 The cyclic case ¶ ↺
      - depends_on proposition A.246 Bounded Borel functional calculus ¶ ↺
    - depends_on theorem 12.38 Existence and uniqueness of the adjoint ¶
      - depends_on definition 12.41 The operator classes ¶ ↺
      - depends_on definition 12.71 Adjoint of a densely defined operator ¶ ↺
      - depends_on proposition 12.39 Algebra of the adjoint; the $C^{\ast}$ identity ¶

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`	→	Orthogonal vectors	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:398
`depends_on`	→	eq:lin-orthogonal-complement	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:398
`depends_on`	←	Orthonormal basis	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:734
`depends_on`	←	Internal orthogonal decomposition	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2392
`depends_on`	←	The complement is always a closed subspace	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:408