lemma A.452 Young and Hölder

open in the book · appendices/A-long-proofs.tex:22204 · p. 3014

Rests on

depends_on definition A.450 The space $W^{1,r}(a,b)$ ¶
- depends_on definition 10.85 Sobolev space ¶
  - depends_on definition 12.2 Hilbert space ¶
    - depends_on definition 5.18 Inner product ¶
      - depends_on definition 4.33 Vector space ¶
    - depends_on definition 6.27 Convergence; Cauchy sequence; completeness ¶
      - depends_on definition 6.24 Metric ¶
    - depends_on equation 5.44 eq:lin-norm-assoc ¶
  - depends_on definition 10.83 Weak solution ¶
    - depends_on definition 17.38 Schwartz space and tempered distributions ¶
    - depends_on definition 10.3 Principal part and principal symbol ¶
      - depends_on definition 10.2 Linear, semilinear, quasilinear ¶
      - depends_on equation 10.2 eq:pde-multi-index ¶
- depends_on theorem 7.42 Fundamental theorem of calculus, I ¶
  - depends_on definition 7.26 Derivative of a function at a point ¶
    - depends_on definition 7.16 Limit ¶
      - depends_on definition 7.2 Absolute value ¶
      - depends_on definition 7.9 Real function ¶
    - depends_on definition 7.9 Real function ¶ ↺
  - depends_on theorem 7.40 Continuous functions are integrable ¶
    - depends_on definition 7.39 Darboux sums and the definite integral ¶
      - depends_on axiom 7.1 Completeness of $\R$ ¶
    - depends_on theorem 7.24 Extreme value theorem ¶
      - depends_on axiom 7.1 Completeness of $\R$ ¶ ↺
      - depends_on proposition 7.22 Sequential characterization ¶
      - depends_on theorem 7.7 Bolzano–Weierstrass ¶
      - proves proof ch:05-real-analysis@proof-9 ¶
    - depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
      - depends_on proposition 7.22 Sequential characterization ¶ ↺
      - depends_on theorem 7.7 Bolzano–Weierstrass ¶ ↺
      - proves proof ch:05-real-analysis@proof-10 ¶
    - proves proof ch:05-real-analysis@proof-23 ¶
  - depends_on theorem 7.24 Extreme value theorem ¶ ↺
  - proves proof ch:05-real-analysis@proof-24 ¶
depends_on theorem 7.38 Taylor's theorem with Lagrange remainder ¶
- depends_on proposition 7.30 Leibniz rule ¶
  - depends_on definition 7.12 Product of functions ¶
    - depends_on definition 7.9 Real function ¶ ↺
  - depends_on equation 7.14 eq:ana-derivh ¶
  - depends_on proposition 7.27 Differentiable implies continuous ¶
    - depends_on definition 7.20 Continuity at a point ¶
      - depends_on definition 7.16 Limit ¶ ↺
      - depends_on equation 7.7 eq:ana-limit-left ¶
      - depends_on equation 7.5 eq:ana-limit-right ¶
    - depends_on definition 7.26 Derivative of a function at a point ¶ ↺
    - depends_on proposition 7.6 Algebra of limits ¶
      - depends_on definition 7.4 Convergence ¶
      - depends_on proposition 7.3 Triangle inequality ¶
      - proves proof ch:05-real-analysis@proof-3 ¶
    - proves proof ch:05-real-analysis@proof-11 ¶
  - depends_on proposition 7.6 Algebra of limits ¶ ↺
  - proves proof ch:05-real-analysis@proof-14 ¶
- depends_on theorem 7.34 Rolle ¶
  - depends_on lemma 7.33 Fermat: interior extremum ¶
    - depends_on definition 7.26 Derivative of a function at a point ¶ ↺
    - depends_on proposition 7.18 Two-sided limit from one-sided limits ¶
      - depends_on definition 7.16 Limit ¶ ↺
      - depends_on equation 7.7 eq:ana-limit-left ¶ ↺
      - depends_on equation 7.5 eq:ana-limit-right ¶ ↺
      - proves proof ch:05-real-analysis@proof-6 ¶
    - proves proof ch:05-real-analysis@proof-17 ¶
  - depends_on theorem 7.24 Extreme value theorem ¶ ↺
  - proves proof ch:05-real-analysis@proof-18 ¶
- proves proof ch:05-real-analysis@proof-22 ¶
proves proof app:A-long-proofs@proof-268 ¶

Supports

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.464 $H_{E}$ is a Hilbert space ¶
- depends_on lemma A.466 The homogeneous Dirichlet problem is trivial ¶
  - depends_on definition A.467 Green function of the shifted problem ¶
    - depends_on lemma A.468 The kernel is well defined, symmetric and Lipschitz ¶
      - depends_on theorem A.470 $T$ is compact ¶
    - depends_on theorem A.469 The Green operator inverts $L_{K}$ ¶
      - depends_on lemma A.472 The pairing identity ¶
      - depends_on theorem A.471 Spectral decomposition and completeness in $L^{2}_{r}$ ¶
  - depends_on theorem A.469 The Green operator inverts $L_{K}$ ¶ ↺
- 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 ¶

Neighborhood

Every logical edge within two steps of this node.

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partly declared
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Edges

type	direction	node	provenance	where
`depends_on`	→	The space $W^{1,r}(a,b)$	declared	appendices/A-long-proofs.tex:22216
`depends_on`	→	Taylor's theorem with Lagrange remainder	declared	appendices/A-long-proofs.tex:22216
`depends_on`	←	Courant–Lebesgue	declared	appendices/A-long-proofs.tex:23982
`depends_on`	←	Poincaré inequality; $B_{K}$ is an inner product	declared	appendices/A-long-proofs.tex:22852
`depends_on`	←	Uniform bound and uniform Hölder continuity	declared	appendices/A-long-proofs.tex:22272
`proves`	←	app:A-long-proofs@proof-268	declared	appendices/A-long-proofs.tex:22219