lemma A.468 The kernel is well defined, symmetric and Lipschitz

open in the book · appendices/A-long-proofs.tex:22974 · p. 3022

Rests on

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 9.2 Linear equation; homogeneity ¶
      - depends_on remark 9.3 Normal form and first-order systems ¶
      - depends_on theorem 9.8 Picard–Lindelöf ¶
      - proves proof ch:07-odes-sturm-liouville@proof-3 ¶
    - depends_on definition A.438 Momentum variable and the accessory system ¶
      - depends_on equation 16.49 eq:calcvar-jacobi ¶
      - depends_on equation 9.3 eq:slt-ode-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 definition A.450 The space $W^{1,r}(a,b)$ ¶
      - depends_on definition 16.75 Rayleigh quotient ¶
    - 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 ¶
      - proves proof app:A-long-proofs@proof-268 ¶
    - 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 definition 3.43 Map ¶
  - depends_on equation 7.14 eq:ana-derivh ¶
  - depends_on proposition 7.6 Algebra of limits ¶
    - depends_on definition 7.4 Convergence ¶
      - depends_on definition 7.2 Absolute value ¶
    - depends_on proposition 7.3 Triangle inequality ¶
      - depends_on definition 7.2 Absolute value ¶ ↺
      - proves proof ch:05-real-analysis@proof-1 ¶
    - 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 definition 7.16 Limit ¶
      - depends_on definition 7.9 Real function ¶ ↺
    - depends_on proposition 7.18 Two-sided limit from one-sided limits ¶
      - depends_on definition 7.16 Limit ¶ ↺
      - depends_on equation 7.7 eq:ana-limit-left ¶
      - depends_on equation 7.5 eq:ana-limit-right ¶
      - proves proof ch:05-real-analysis@proof-6 ¶
    - 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 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 ¶
  - proves proof ch:05-real-analysis@proof-18 ¶
- proves proof ch:05-real-analysis@proof-19 ¶
proves proof app:A-long-proofs@proof-277 ¶

Supports

depends_on theorem A.470 $T$ is compact ¶
- 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`	→	Green function of the shifted problem	declared	appendices/A-long-proofs.tex:22986
`depends_on`	→	Mean value theorem	declared	appendices/A-long-proofs.tex:22986
`depends_on`	←	$T$ is compact	declared	appendices/A-long-proofs.tex:23151
`proves`	←	app:A-long-proofs@proof-277	declared	appendices/A-long-proofs.tex:22989