definition 3.53 Preimage

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Supports

• depends_on definition 6.6 Continuous map ¶
  • depends_on definition A.361 Covering map ¶
    • depends_on lemma A.366 Unique path lifting ¶
      • depends_on lemma A.367 Homotopy lifting ¶
    • depends_on lemma A.369 $\Phi$ is a two-sheeted covering ¶
    • depends_on theorem A.364 Covering homomorphisms ¶
    • depends_on theorem A.363 Monodromy ¶
      • depends_on theorem A.364 Covering homomorphisms ¶ ↺
  • depends_on definition 12.64 Strongly continuous one-parameter unitary group ¶
    • depends_on definition A.580 Weyl operator ¶
      • 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.581 Composition law ¶
        • depends_on lemma A.582 Joint strong continuity ¶
        • depends_on lemma A.590 The Gram matrix is universal ¶
    • depends_on definition 12.109 Weyl system ¶
      • depends_on corollary 12.112 Commutator of the momentum with a function of the position ¶
      • depends_on definition A.580 Weyl operator ¶ ↺
      • 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 proposition 12.111 The Weyl relation is a covariance statement ¶
        • depends_on corollary 12.112 Commutator of the momentum with a function of the position ¶ ↺
      • depends_on theorem A.579 Stone–von Neumann ¶ ↺
      • depends_on theorem 12.114 Stone–von Neumann ¶
    • 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.581 Composition law ¶ ↺
    • depends_on lemma A.582 Joint strong continuity ¶ ↺
    • depends_on proposition A.256 Density ¶ ↺
    • 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 12.67 The generator is symmetric, and generates the motion ¶ ↺
    • depends_on theorem A.253 Stone ¶
      • depends_on proposition A.263 The two constructions are inverse ¶
    • depends_on theorem 12.66 Stone ¶ ↺
    • depends_on theorem 25.34 Stone–von Neumann ¶ ↺
  • depends_on definition 6.7 Homeomorphism ¶
    • depends_on definition 13.50 Diffeomorphism ¶
    • depends_on definition 13.53 Immersion, submersion, embedding ¶
      • depends_on definition 13.54 Embedded submanifold ¶
        • depends_on corollary 13.64 The image of a constant-rank map, locally ¶
        • depends_on definition 13.130 Distribution; involutive; integrable ¶
        • depends_on definition 13.58 Hypersurface ¶
        • depends_on proposition A.534 Every orbit is an embedded copy of $G$ ¶
        • depends_on theorem 13.67 Quotient manifold theorem ¶
        • depends_on theorem 13.62 Regular value theorem in codimension $k$ ¶
      • depends_on example 13.55 An injective immersion that is not an embedding ¶
      • depends_on lemma A.533 The orbit map has constant rank $d$ ¶
        • depends_on proposition A.534 Every orbit is an embedded copy of $G$ ¶ ↺
        • depends_on proposition A.535 Existence of a slice ¶
      • depends_on lemma A.538 Submersions have smooth local sections ¶
        • depends_on proposition A.539 Universal property ¶
      • depends_on proposition A.534 Every orbit is an embedded copy of $G$ ¶ ↺
      • depends_on theorem A.537 The smooth structure on the orbit space ¶
    • depends_on lemma A.536 A slice is a chart domain downstairs ¶
      • depends_on theorem A.537 The smooth structure on the orbit space ¶ ↺
    • depends_on theorem 32.15 Hartman–Grobman, restated from Part II ¶
      • depends_on definition 32.22 Bifurcation ¶
        • depends_on proposition 32.23 Saddle-node bifurcation ¶
        • depends_on proposition 32.24 Transcritical and pitchfork bifurcations ¶
        • depends_on theorem 32.25 Hopf bifurcation, quoted ¶
      • depends_on example 32.18 A linear centre that is really a stable focus ¶
    • depends_on theorem 9.32 Hartman–Grobman; quoted ¶
      • depends_on theorem 32.15 Hartman–Grobman, restated from Part II ¶ ↺
  • depends_on definition 6.15 Path-connected space ¶
    • depends_on definition A.362 Path homotopy and the fundamental group ¶
      • depends_on lemma A.368 The concatenation rules ¶
      • depends_on theorem A.363 Monodromy ¶ ↺
    • depends_on definition 6.17 Simply connected space ¶
      • depends_on corollary 8.15 Deformation of contours ¶
        • depends_on lemma 106.1 The short-time kernel ¶
        • depends_on theorem 8.16 Cauchy integral formula ¶
        • depends_on theorem 8.21 Laurent expansion ¶
        • depends_on theorem 8.24 Residue theorem ¶
      • depends_on definition A.362 Path homotopy and the fundamental group ¶ ↺
      • depends_on definition 16.62 Field of extremals; slope function ¶
        • depends_on definition 23.52 Lagrangian family; caustic ¶
        • depends_on remark 23.8 The action as a function, not a functional ¶
        • depends_on theorem 16.63 Hilbert's invariant integral ¶
        • depends_on theorem 16.64 Weierstrass' sufficient condition ¶
      • depends_on example 6.18 ex:top-simply-connected ¶
        • depends_on lemma 14.34 The $n$-sphere is simply connected for $n \ge 2$ ¶
      • depends_on lemma 14.34 The $n$-sphere is simply connected for $n \ge 2$ ¶ ↺
      • depends_on proposition 32.33 Bendixson's negative criterion ¶
      • depends_on proposition 14.31 $\SO(3,\R)$ is not simply connected ¶
        • depends_on remark 29.12 The coordinate singularity, and what it is not ¶
      • depends_on proposition 9.36 Bendixson–Dulac negative criterion ¶
      • depends_on proposition 6.23 The punctured plane is not simply connected ¶
    • depends_on proposition 6.16 prop:top-path-implies-connected ¶
  • depends_on definition 6.17 Simply connected space ¶ ↺
  • 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 lemma A.531 The projection is open ¶
    • depends_on lemma A.536 A slice is a chart domain downstairs ¶ ↺
    • depends_on proposition A.532 Separation and countability of the quotient ¶
      • depends_on remark A.542 Where each hypothesis is spent ¶
      • depends_on theorem A.537 The smooth structure on the orbit space ¶ ↺
  • depends_on proposition 6.10 Continuous images of compact sets ¶
    • depends_on lemma A.530 Two elementary facts about compactness ¶
      • depends_on proposition A.532 Separation and countability of the quotient ¶ ↺
  • 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 notation 4.13 The opposite map is not the inverse map ¶
• depends_on proposition 3.54 Preimages respect the set operations ¶