definition 6.24 Metric

open in the book · parts/02-mathematical-methods/04-topology.tex:481 · p. 198

ground object -- no derivation owed

Rests on

depends_on definition 3.43 Map ¶
- depends_on definition 3.24 Quantifiers ¶
  - depends_on definition 3.23 Predicate ¶
    - depends_on definition 3.1 Logical proposition ¶
    - depends_on definition 3.38 Ordered pair and Cartesian product ¶
- depends_on definition 3.28 Set ¶
depends_on definition 3.38 Ordered pair and Cartesian product ¶ ↺

Supports

depends_on definition 7.140 Hausdorff measure ¶
- depends_on definition 7.142 Box-counting dimension ¶
  - depends_on definition 32.79 Box-counting dimension, restated from Part II ¶
    - depends_on definition 32.81 Correlation dimension ¶
    - depends_on example 32.80 The middle-thirds Cantor set ¶
  - depends_on proposition 7.143 The Hausdorff dimension never exceeds the box dimension ¶
  - depends_on remark 7.144 What is developed here, and what is not ¶
- depends_on proposition 7.141 The critical exponent, and the Hausdorff dimension ¶
  - depends_on proposition 7.143 The Hausdorff dimension never exceeds the box dimension ¶ ↺
- depends_on remark 7.144 What is developed here, and what is not ¶ ↺
depends_on definition 16.12 The two norms; weak and strong neighbourhoods ¶
- depends_on definition 16.13 Weak and strong extrema ¶
  - depends_on example 16.14 A weak minimum that is not strong ¶
  - depends_on proposition 16.16 Stationarity is necessary ¶
    - depends_on theorem 16.22 Euler–Lagrange ¶
      - depends_on definition 16.62 Field of extremals; slope function ¶
      - depends_on phenomenon 30.52 Euler buckling ¶
      - depends_on proposition 29.39 Steady precession ¶
      - depends_on theorem 16.35 System of Euler–Lagrange equations ¶
      - depends_on theorem 16.31 Natural boundary condition ¶
    - depends_on theorem 16.51 Legendre's necessary condition ¶
      - depends_on corollary 16.65 Sufficiency for a weak minimum ¶
      - depends_on theorem 16.55 Jacobi's necessary condition ¶
  - depends_on theorem 16.60 Weierstrass' necessary condition ¶
depends_on definition 5.22 Metric associated with a norm ¶
depends_on definition 10.24 Well-posed problem ¶
- depends_on corollary A.152 Uniqueness ¶
- depends_on corollary 10.76 Uniqueness and stability for the Dirichlet problem ¶
- depends_on corollary 10.81 Uniqueness for the heat equation ¶
- depends_on definition 30.22 Boundary conditions of elastostatics ¶
  - depends_on proposition 30.50 Love waves need a slow surface layer ¶
  - depends_on proposition 30.48 Rayleigh's secular equation ¶
    - depends_on proposition 30.49 The Rayleigh speed of a Poisson solid ¶
      - depends_on phenomenon 30.47 The three seismic phases ¶
  - depends_on proposition 30.23 Virtual work; the weak form ¶
    - depends_on phenomenon 30.70 Hertzian contact ¶
      - depends_on definition A.698 The Hertz contact problem ¶
      - depends_on example A.703 The numbers of the chapter, recomputed ¶
      - depends_on example 30.71 Contact stresses are enormous ¶
    - depends_on proposition 30.34 Minimum of the potential energy ¶
      - depends_on example 30.35 From single crystals to the isotropic aggregate ¶
    - depends_on remark 30.69 Saint-Venant's principle, and its exact status ¶
- depends_on proposition 10.25 The backward heat problem is ill posed ¶
- depends_on theorem 44.45 Local existence and uniqueness; Choquet-Bruhat; imported ¶
  - depends_on theorem 44.46 Maximal globally hyperbolic development; Choquet-Bruhat and Geroch; imported ¶
depends_on definition 6.27 Convergence; Cauchy sequence; completeness ¶
- 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 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 12.2 Hilbert space ¶
  - 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 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.453 Uniform bound and uniform Hölder continuity ¶
      - depends_on lemma A.452 Young and Hölder ¶
    - 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 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 12.10 $\ell^{2}$ 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 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 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 lemma A.257 Integrated form of the equation of motion ¶
      - depends_on proposition A.258 The generator is closed ¶
    - depends_on lemma A.583 Absolutely convergent operator-valued integrals ¶
      - depends_on definition A.585 The Gaussian average of the Weyl operators ¶
  - 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 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 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 12.28 Convergence criterion for orthogonal series ¶
    - 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 theorem A.229 Hilbert–Schmidt ¶
      - depends_on theorem 12.44 Hilbert–Schmidt: compact self-adjoint operators ¶
      - depends_on theorem 12.33 Every separable Hilbert space is $\ell^{2}$ ¶ ↺
  - 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 ¶
      - 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 ¶
  - … 1 more
- depends_on definition 6.29 Sequential compactness ¶
  - depends_on lemma 6.30 Lebesgue number ¶
    - depends_on theorem 6.31 Compactness and sequential compactness ¶
      - depends_on lemma A.230 Sequential characterisation ¶
      - depends_on lemma A.488 Small chords cut off small arcs ¶
  - depends_on theorem 6.31 Compactness and sequential compactness ¶ ↺
- depends_on proposition 12.8 Absolutely convergent series test ¶
  - 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 A.241 The polynomial calculus is isometric ¶
    - depends_on proposition 12.65 Exponential of a bounded self-adjoint operator ¶
      - depends_on theorem 12.66 Stone ¶
    - 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 definition 12.79 Deficiency subspaces and indices ¶
      - depends_on proposition A.241 The polynomial calculus is isometric ¶ ↺
      - depends_on proposition 12.65 Exponential of a bounded self-adjoint operator ¶ ↺
      - depends_on proposition 12.42 Elementary consequences ¶
      - depends_on proposition 12.43 Norm of a self-adjoint operator ¶
      - depends_on proposition 12.96 Operators on a tensor product ¶ ↺
    - 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 12.38 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 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 theorem 12.12 Riesz–Fischer ¶ ↺
depends_on definition 6.26 Open ball; metric topology ¶
- depends_on definition A.76 Star-shaped set ¶
  - depends_on lemma A.77 Poincaré lemma, converse form ¶
- depends_on lemma 6.30 Lebesgue number ¶ ↺
- depends_on proposition 6.28 $\varepsilon$–$\delta$ characterization ¶
  - 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 definition 6.29 Sequential compactness ¶ ↺
depends_on lemma A.505 A continuous partition of unity ¶
- depends_on lemma A.506 Locality ¶
  - depends_on proposition A.511 The substitution property is universal ¶
    - depends_on remark A.516 The hypotheses of the global form ¶
    - depends_on theorem A.513 Change of variables ¶
      - depends_on corollary A.514 Degeneracy on a negligible set ¶
      - depends_on remark A.516 The hypotheses of the global form ¶ ↺

Neighborhood

Every logical edge within two steps of this node.

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partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Map	declared	parts/02-mathematical-methods/04-topology.tex:500
`depends_on`	→	Ordered pair and Cartesian product	declared	parts/02-mathematical-methods/04-topology.tex:500
`depends_on`	←	Hausdorff measure	declared	parts/02-mathematical-methods/05-real-analysis.tex:4775
`depends_on`	←	The two norms; weak and strong neighbourhoods	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:412
`depends_on`	←	Metric associated with a norm	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:1170
`depends_on`	←	Well-posed problem	declared	parts/02-mathematical-methods/08-pdes.tex:661
`depends_on`	←	Convergence; Cauchy sequence; completeness	declared	parts/02-mathematical-methods/04-topology.tex:532
`depends_on`	←	Open ball; metric topology	declared	parts/02-mathematical-methods/04-topology.tex:520
`depends_on`	←	Sequential compactness	declared	parts/02-mathematical-methods/04-topology.tex:584
`depends_on`	←	A continuous partition of unity	declared	appendices/A-long-proofs.tex:24717