theorem A.455 Compact embedding of $W^{1,r}(a,b)$ into the continuous functions

open in the book · appendices/A-long-proofs.tex:22370 · p. 3015

Rests on

depends_on lemma A.454 Arzelà–Ascoli on an interval ¶
- depends_on theorem 7.7 Bolzano–Weierstrass ¶
  - depends_on corollary A.45 Archimedean property and density of $\Q$ ¶
    - depends_on definition A.36 Cut ¶
    - depends_on theorem A.40 Least-upper-bound property ¶
      - depends_on definition A.36 Cut ¶ ↺
      - depends_on definition A.38 Order on $\R$ ¶
      - proves proof app:A-long-proofs@proof-29 ¶
    - proves proof app:A-long-proofs@proof-32 ¶
  - depends_on corollary A.46 Monotone convergence ¶
    - depends_on theorem A.40 Least-upper-bound property ¶ ↺
    - proves proof app:A-long-proofs@proof-33 ¶
  - proves proof ch:05-real-analysis@proof-4 ¶
- depends_on theorem 7.8 Cauchy criterion ¶
  - depends_on corollary A.47 Cauchy completeness ¶
    - depends_on corollary A.46 Monotone convergence ¶ ↺
    - depends_on theorem A.40 Least-upper-bound property ¶ ↺
    - proves proof app:A-long-proofs@proof-34 ¶
  - depends_on definition 7.4 Convergence ¶
    - depends_on definition 7.2 Absolute value ¶
  - proves proof ch:05-real-analysis@proof-5 ¶
- proves proof app:A-long-proofs@proof-270 ¶
depends_on lemma A.453 Uniform bound and uniform Hölder continuity ¶
- 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 6.27 Convergence; Cauchy sequence; completeness ¶
      - 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 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 7.9 Real function ¶
    - depends_on theorem 7.40 Continuous functions are integrable ¶
      - depends_on definition 7.39 Darboux sums and the definite integral ¶
      - depends_on theorem 7.24 Extreme value theorem ¶
      - depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
      - 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 equation 7.14 eq:ana-derivh ¶
      - depends_on proposition 7.27 Differentiable implies continuous ¶
      - 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 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 ¶
proves proof app:A-long-proofs@proof-271 ¶

Supports

depends_on theorem A.456 Convexity in $q$ implies weak lower semicontinuity ¶

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
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declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Arzelà–Ascoli on an interval	declared	appendices/A-long-proofs.tex:22385
`depends_on`	→	Uniform bound and uniform Hölder continuity	declared	appendices/A-long-proofs.tex:22385
`depends_on`	←	Convexity in $q$ implies weak lower semicontinuity	declared	appendices/A-long-proofs.tex:22444
`proves`	←	app:A-long-proofs@proof-271	declared	appendices/A-long-proofs.tex:22389