lemma A.463 Poincaré inequality; $B_{K}$ is an inner product

open in the book · appendices/A-long-proofs.tex:22833 · p. 3021

Rests on

depends_on definition A.462 The weighted space and the energy space ¶
- 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 definition 16.75 Rayleigh quotient ¶
  - depends_on definition 9.54 Sturm–Liouville differential operator ¶
  - depends_on equation 16.42 eq:calcvar-sl-from-isoperimetric ¶
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-274 ¶

Supports

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.471 Spectral decomposition and completeness in $L^{2}_{r}$ ¶
  - 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 $L_{K}$ ¶ ↺

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`	→	The weighted space and the energy space	declared	appendices/A-long-proofs.tex:22852
`depends_on`	→	Young and Hölder	declared	appendices/A-long-proofs.tex:22852
`depends_on`	←	$H_{E}$ is a Hilbert space	declared	appendices/A-long-proofs.tex:22882
`depends_on`	←	The homogeneous Dirichlet problem is trivial	declared	appendices/A-long-proofs.tex:22932
`depends_on`	←	The Green operator inverts $L_{K}$	declared	appendices/A-long-proofs.tex:23039
`proves`	←	app:A-long-proofs@proof-274	declared	appendices/A-long-proofs.tex:22856