definition 5.19 Norm

open in the book · parts/02-mathematical-methods/03-linear-algebra-representations.tex:975 · p. 108

ground object -- no derivation owed

Rests on

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 4.2 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 4.3 Algebraic structure ¶ ↺
  - depends_on definition 4.20 Monoid ¶
    - depends_on definition 4.9 Neutral element ¶ ↺
    - depends_on definition 4.18 Semigroup ¶
      - depends_on definition 4.6 Associativity ¶
      - depends_on definition 4.4 Internal binary operation; magma ¶ ↺
  - depends_on definition 4.30 Ring ¶
    - depends_on definition 4.8 Commutativity; abelian structure ¶ ↺
    - depends_on definition 4.28 Distributivity ¶
      - depends_on definition 4.3 Algebraic structure ¶ ↺
    - depends_on definition 4.21 Group ¶
      - depends_on definition 4.11 Inverse element ¶ ↺
      - depends_on definition 4.20 Monoid ¶ ↺
    - 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 ¶ ↺

Supports

depends_on definition 12.35 Bounded operator; operator norm ¶
- depends_on definition 12.45 Continuous linear functional; the dual ¶
  - depends_on corollary 12.47 $\mathcal{H}$ is its own dual, antilinearly ¶
    - depends_on definition 12.103 Gelfand triple ¶
      - depends_on definition A.278 Countably Hilbert nuclear space ¶
      - depends_on definition 12.105 Generalized eigenvector ¶
      - depends_on example 12.104 The Schwartz triple ¶
      - depends_on theorem A.279 Gelfand–Maurin ¶
      - depends_on theorem 12.107 Nuclear spectral theorem ¶
  - depends_on definition 12.103 Gelfand triple ¶ ↺
  - depends_on theorem 12.46 Riesz representation ¶
    - depends_on corollary 12.47 $\mathcal{H}$ is its own dual, antilinearly ¶ ↺
    - 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 ¶
- depends_on definition 12.88 Reducing subspace ¶
  - depends_on definition 12.90 Self-adjoint family; commutant; irreducibility ¶
    - depends_on proposition A.589 Cyclic subspaces and the rank of the average ¶
      - depends_on lemma A.590 The Gram matrix is universal ¶
      - depends_on proposition A.591 Any two irreducible Weyl systems are equivalent ¶
    - depends_on theorem A.579 Stone–von Neumann ¶
    - depends_on theorem 12.91 Schur's lemma, commutant form ¶
      - depends_on proposition A.591 Any two irreducible Weyl systems are equivalent ¶ ↺
      - depends_on theorem 12.114 Stone–von Neumann ¶
    - depends_on theorem 12.114 Stone–von Neumann ¶ ↺
  - depends_on proposition 12.89 Reduction is commutation ¶
    - depends_on proposition A.589 Cyclic subspaces and the rank of the average ¶ ↺
    - depends_on theorem 12.91 Schur's lemma, commutant form ¶ ↺
- depends_on definition 12.49 Resolvent set; spectrum ¶
  - 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 lemma A.240 Spectral mapping for polynomials ¶
    - depends_on proposition A.241 The polynomial calculus is isometric ¶
      - depends_on proposition A.243 Continuous functional calculus ¶
  - depends_on lemma A.239 The norm of a self-adjoint operator lies in its spectrum ¶
    - depends_on proposition A.241 The polynomial calculus is isometric ¶ ↺
  - depends_on proposition 12.52 Neumann series; the spectrum is bounded ¶
    - depends_on lemma A.239 The norm of a self-adjoint operator lies in its spectrum ¶ ↺
    - depends_on proposition A.261 Spectral theorem for a unitary operator ¶ ↺
    - depends_on proposition 12.53 The resolvent set is open, the resolvent analytic ¶
      - depends_on theorem 12.54 The spectrum is compact and non-empty ¶
    - depends_on theorem 12.54 The spectrum is compact and non-empty ¶ ↺
- depends_on proposition 12.37 $\mathcal{B}(\mathcal{H})$ is a Banach algebra ¶
  - depends_on lemma A.240 Spectral mapping for polynomials ¶ ↺
  - depends_on proposition 12.65 Exponential of a bounded self-adjoint operator ¶
    - depends_on theorem 12.66 Stone ¶
      - depends_on corollary 12.112 Commutator of the momentum with a function of the position ¶
      - depends_on definition 12.109 Weyl system ¶
      - depends_on proposition 12.67 The generator is symmetric, and generates the motion ¶
      - depends_on proposition 12.111 The Weyl relation is a covariance statement ¶
      - depends_on theorem 25.34 Stone–von Neumann ¶
  - depends_on proposition 12.61 Uniqueness of the continuous functional calculus ¶
    - depends_on proposition A.243 Continuous functional calculus ¶ ↺
  - depends_on proposition 12.39 Algebra of the adjoint; the $C^{\ast}$ identity ¶ ↺
  - depends_on proposition 12.52 Neumann series; the spectrum is bounded ¶ ↺
  - depends_on theorem 12.38 Existence and uniqueness of the adjoint ¶ ↺
  - depends_on theorem 12.75 The canonical commutation relation admits no bounded solution ¶
    - depends_on corollary 12.76 Position and momentum are unbounded, and cannot be everywhere defined ¶
    - depends_on corollary 25.30 The relation cannot be realized by matrices ¶
    - depends_on definition 12.109 Weyl system ¶ ↺
- depends_on proposition 12.36 Boundedness is continuity ¶
  - depends_on theorem 12.74 Hellinger–Toeplitz ¶
    - depends_on corollary 12.76 Position and momentum are unbounded, and cannot be everywhere defined ¶ ↺
- depends_on theorem 12.75 The canonical commutation relation admits no bounded solution ¶ ↺
depends_on definition 5.22 Metric associated with a norm ¶
depends_on definition 5.25 Unit vector ¶
- depends_on definition 5.27 Orthonormal basis ¶
  - depends_on definition 12.26 Orthonormal system; Fourier coefficients ¶
    - depends_on definition 12.29 Orthonormal basis ¶
      - depends_on definition 12.86 Internal orthogonal decomposition ¶
    - depends_on proposition 12.27 Best approximation and Bessel's inequality ¶
      - depends_on theorem 12.30 Completeness, expansion, Parseval ¶
    - depends_on proposition 12.28 Convergence criterion for orthogonal series ¶
      - depends_on theorem 12.30 Completeness, expansion, Parseval ¶ ↺
  - depends_on definition 5.32 Levi–Civita symbol; cross product ¶
    - depends_on definition 5.131 Isotropic Cartesian tensor ¶
      - depends_on definition A.672 The octahedral rotation group ¶
      - depends_on lemma 5.132 Parity constraint ¶
    - depends_on lemma 5.34 Contraction of two Levi–Civita symbols ¶
      - depends_on proposition 5.35 The identities of the vector algebra of $\R^{3}$ ¶
    - depends_on lemma 5.33 Determinant through the Levi–Civita symbol ¶
      - depends_on proposition 5.35 The identities of the vector algebra of $\R^{3}$ ¶ ↺
      - depends_on theorem 5.133 Isotropic Cartesian tensors of rank at most four ¶
  - depends_on proposition 12.23 Gram–Schmidt in a Hilbert space ¶
    - depends_on corollary 12.24 Separable spaces have countable orthonormal families ¶
      - depends_on theorem 12.33 Every separable Hilbert space is $\ell^{2}$ ¶
    - depends_on example 12.25 Orthogonal polynomials ¶
    - 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 proposition 5.28 Gram–Schmidt ¶
    - depends_on corollary 5.29 Orthogonal decomposition ¶
      - depends_on proposition 5.153 prop:rep-unitary-completely-reducible ¶
      - depends_on theorem 5.43 The four fundamental subspaces ¶
    - depends_on proposition 12.23 Gram–Schmidt in a Hilbert space ¶ ↺
    - depends_on proposition 5.42 The adjoint exists, is unique, and is linear ¶
      - depends_on corollary 5.82 Spectral theorem for a normal operator ¶
    - depends_on proposition 5.155 Averaging trick ¶
      - depends_on theorem 101.62 The mixing matrix is unitary ¶
    - depends_on theorem 5.86 Simultaneous diagonalization of a definite pencil ¶
      - depends_on corollary 5.89 Rayleigh–Ritz ¶
      - depends_on proposition 5.88 Rayleigh quotient of a definite pencil ¶
    - depends_on theorem 5.84 Spectral theorem for a real symmetric operator ¶
      - depends_on corollary 5.91 The operator norm of a real array ¶
      - depends_on lemma A.610 Block positivity ¶
      - depends_on lemma A.601 Several variables, complex symmetric matrix ¶
      - depends_on lemma A.600 The real part of an inverse ¶
      - depends_on proposition 5.90 Polar decomposition ¶
      - depends_on proposition 5.94 Principal axes of a real quadratic form ¶
      - depends_on proposition 5.92 The eigenvalues do not control the norm ¶
      - depends_on theorem 5.86 Simultaneous diagonalization of a definite pencil ¶ ↺
depends_on definition 9.21 Matrix exponential ¶
- depends_on proposition 9.22 The exponential and its derivative ¶
  - depends_on theorem 9.23 Solution of a constant-coefficient system ¶
    - depends_on corollary 9.25 Normal modes of a diagonalizable system ¶
      - depends_on corollary 9.27 Linear stability ¶
      - depends_on proposition 9.29 The planar classification ¶
    - depends_on proposition 9.87 The distinguished limit, and which end carries the layer ¶
      - depends_on proposition 9.88 The model problem, and the error of the composite ¶
    - depends_on proposition 9.31 Lyapunov's first method: asymptotic stability ¶
    - depends_on proposition 9.43 The unmodulated line, and the feet of the tongues ¶
    - depends_on theorem 9.32 Hartman–Grobman; quoted ¶
      - depends_on theorem 32.15 Hartman–Grobman, restated from Part II ¶
    - depends_on theorem 9.26 Structure of the solutions ¶
      - depends_on corollary 9.27 Linear stability ¶ ↺
depends_on proposition 12.8 Absolutely convergent series test ¶
- depends_on proposition 12.37 $\mathcal{B}(\mathcal{H})$ is a Banach algebra ¶ ↺
- depends_on theorem 12.12 Riesz–Fischer ¶
  - depends_on lemma A.464 $H_{E}$ is a Hilbert space ¶

Neighborhood

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Edges

type	direction	node	provenance	where
`depends_on`	→	Vector space	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:1001
`depends_on`	←	Bounded operator; operator norm	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:894
`depends_on`	←	Metric associated with a norm	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:1170
`depends_on`	←	Unit vector	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:1255
`depends_on`	←	Matrix exponential	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:738
`depends_on`	←	Absolutely convergent series test	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:208