definition A.462 The weighted space and the energy space

open in the book · appendices/A-long-proofs.tex:22816 · p. 3020

ground object -- no derivation owed

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 definition 16.75 Rayleigh quotient ¶
- depends_on definition 9.54 Sturm–Liouville differential operator ¶
- depends_on equation 16.42 eq:calcvar-sl-from-isoperimetric ¶

Supports

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 $L_{K}$ ¶ ↺

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 space $W^{1,r}(a,b)$	declared	appendices/A-long-proofs.tex:22830
`depends_on`	→	Rayleigh quotient	declared	appendices/A-long-proofs.tex:22830
`depends_on`	←	Poincaré inequality; $B_{K}$ is an inner product	declared	appendices/A-long-proofs.tex:22852