theorem A.470 $T$ is compact

open in the book · appendices/A-long-proofs.tex:23147 · p. 3024

Rests on

depends_on lemma A.468 The kernel is well defined, symmetric and Lipschitz ¶
- depends_on definition A.467 Green function of the shifted problem ¶
  - depends_on lemma A.442 The Wronskian of the accessory equation is constant ¶
    - depends_on lemma A.439 Existence, uniqueness, and the structure of the zeros ¶
      - depends_on corollary 9.9 Linear equations: existence on the whole interval ¶
      - depends_on definition A.438 Momentum variable and the accessory system ¶
      - proves proof app:A-long-proofs@proof-261 ¶
    - depends_on lemma 9.56 Lagrange identity ¶
      - depends_on definition 9.54 Sturm–Liouville differential operator ¶
      - proves proof ch:07-odes-sturm-liouville@proof-28 ¶
    - proves proof app:A-long-proofs@proof-264 ¶
  - depends_on lemma A.466 The homogeneous Dirichlet problem is trivial ¶
    - depends_on equation A.764 eq:app-sl-completeness-operator ¶
    - depends_on lemma A.463 Poincaré inequality; $B_{K}$ is an inner product ¶
      - depends_on definition A.462 The weighted space and the energy space ¶
      - depends_on lemma A.452 Young and Hölder ¶
      - proves proof app:A-long-proofs@proof-274 ¶
    - proves proof app:A-long-proofs@proof-276 ¶
- depends_on theorem 7.35 Mean value theorem ¶
  - depends_on proposition 7.29 Linearity ¶
    - depends_on definition 7.11 Sum of functions ¶
      - depends_on definition 7.9 Real function ¶
    - depends_on equation 7.14 eq:ana-derivh ¶
    - depends_on proposition 7.6 Algebra of limits ¶
      - depends_on definition 7.4 Convergence ¶
      - depends_on proposition 7.3 Triangle inequality ¶
      - proves proof ch:05-real-analysis@proof-3 ¶
    - proves proof ch:05-real-analysis@proof-13 ¶
  - 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 ¶
      - 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 ¶
    - proves proof ch:05-real-analysis@proof-18 ¶
  - proves proof ch:05-real-analysis@proof-19 ¶
- proves proof app:A-long-proofs@proof-277 ¶
depends_on lemma A.454 Arzelà–Ascoli on an interval ¶
- depends_on theorem 7.7 Bolzano–Weierstrass ¶ ↺
- 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-33 ¶
    - depends_on theorem A.40 Least-upper-bound property ¶ ↺
    - proves proof app:A-long-proofs@proof-34 ¶
  - depends_on definition 7.4 Convergence ¶ ↺
  - proves proof ch:05-real-analysis@proof-5 ¶
- proves proof app:A-long-proofs@proof-270 ¶
proves proof app:A-long-proofs@proof-279 ¶

Supports

depends_on theorem A.471 Spectral decomposition and completeness in $L^{2}_{r}$ ¶
- depends_on lemma A.472 The pairing identity ¶

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 kernel is well defined, symmetric and Lipschitz	declared	appendices/A-long-proofs.tex:23151
`depends_on`	→	Arzelà–Ascoli on an interval	declared	appendices/A-long-proofs.tex:23151
`depends_on`	←	Spectral decomposition and completeness in $L^{2}_{r}$	declared	appendices/A-long-proofs.tex:23190
`proves`	←	app:A-long-proofs@proof-279	declared	appendices/A-long-proofs.tex:23154