definition A.450 The space $W^{1,r}(a,b)$

open in the book · appendices/A-long-proofs.tex:22166 · p. 3013

ground object -- no derivation owed

Rests on

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 4.8 Commutativity; abelian structure ¶
      - depends_on definition 4.32 Field ¶
      - depends_on definition 4.31 Module ¶
  - depends_on definition 6.27 Convergence; Cauchy sequence; completeness ¶
    - depends_on definition 6.24 Metric ¶
      - depends_on definition 3.43 Map ¶
      - depends_on definition 3.38 Ordered pair and Cartesian product ¶
  - 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 definition 10.1 Partial differential equation; order ¶
    - 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 3.43 Map ¶ ↺
  - 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 definition 7.20 Continuity at a point ¶
      - depends_on definition 7.4 Convergence ¶
      - proves proof ch:05-real-analysis@proof-7 ¶
    - depends_on theorem 7.7 Bolzano–Weierstrass ¶
      - depends_on corollary A.45 Archimedean property and density of $\Q$ ¶
      - depends_on corollary A.46 Monotone convergence ¶
      - proves proof ch:05-real-analysis@proof-4 ¶
    - 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 ¶

Supports

depends_on definition A.462 The weighted space and the energy space ¶
- 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.469 The Green operator inverts $L_{K}$ ¶
    - 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 ¶
depends_on lemma A.452 Young and Hölder ¶
- 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.453 Uniform bound and uniform Hölder continuity ¶ ↺

Neighborhood

Every logical edge within two steps of this node.

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

Edges

type	direction	node	provenance	where
`depends_on`	→	Sobolev space	declared	appendices/A-long-proofs.tex:22181
`depends_on`	→	Fundamental theorem of calculus, I	declared	appendices/A-long-proofs.tex:22181
`depends_on`	←	The weighted space and the energy space	declared	appendices/A-long-proofs.tex:22830
`depends_on`	←	Uniform bound and uniform Hölder continuity	declared	appendices/A-long-proofs.tex:22272
`depends_on`	←	Young and Hölder	declared	appendices/A-long-proofs.tex:22216