theorem 10.88 Lax–Milgram

open in the book · parts/02-mathematical-methods/08-pdes.tex:2339 · p. 365

Rests on

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 4.4 Internal binary operation; magma ¶
    - depends_on definition 4.32 Field ¶
      - depends_on definition 4.11 Inverse element ¶
      - depends_on definition 4.20 Monoid ¶
      - depends_on definition 4.30 Ring ¶
    - depends_on definition 4.31 Module ¶
      - depends_on definition 4.29 Action ¶
      - depends_on definition 4.28 Distributivity ¶
      - depends_on definition 4.21 Group ¶
      - depends_on definition 4.30 Ring ¶ ↺
- depends_on definition 6.27 Convergence; Cauchy sequence; completeness ¶
  - depends_on definition 6.24 Metric ¶
    - depends_on definition 3.43 Map ¶
      - depends_on definition 3.24 Quantifiers ¶
      - depends_on definition 3.28 Set ¶
    - depends_on definition 3.38 Ordered pair and Cartesian product ¶
- depends_on equation 5.44 eq:lin-norm-assoc ¶
depends_on definition 10.87 Weak form of an elliptic problem ¶
- depends_on definition 10.85 Sobolev space ¶
  - depends_on definition 12.2 Hilbert space ¶ ↺
  - 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 definition 10.83 Weak solution ¶ ↺
proves proof ch:08-pdes@proof-36 ¶

Supports

depends_on corollary 10.89 Existence for the elliptic problem ¶
depends_on proposition 10.91 Dirichlet principle ¶

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`	→	Hilbert space	declared	parts/02-mathematical-methods/08-pdes.tex:2350
`depends_on`	→	Weak form of an elliptic problem	declared	parts/02-mathematical-methods/08-pdes.tex:2350
`depends_on`	←	Existence for the elliptic problem	declared	parts/02-mathematical-methods/08-pdes.tex:2388
`depends_on`	←	Dirichlet principle	declared	parts/02-mathematical-methods/08-pdes.tex:2415
`proves`	←	ch:08-pdes@proof-36	declared	parts/02-mathematical-methods/08-pdes.tex:2353