lemma A.453 Uniform bound and uniform Hölder continuity

open in the book · appendices/A-long-proofs.tex:22260 · 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 lemma A.452 Young and Hölder ¶
- depends_on definition A.450 The space $W^{1,r}(a,b)$ ¶ ↺
- 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 7.26 Derivative of a function at a point ¶ ↺
      - depends_on proposition 7.6 Algebra of limits ¶
      - 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 ¶
      - 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 ¶
proves proof app:A-long-proofs@proof-269 ¶

Supports

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

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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:22272
`depends_on`	→	Young and Hölder	declared	appendices/A-long-proofs.tex:22272
`depends_on`	←	Compact embedding of $W^{1,r}(a,b)$ into the continuous functions	declared	appendices/A-long-proofs.tex:22385
`proves`	←	app:A-long-proofs@proof-269	declared	appendices/A-long-proofs.tex:22275