theorem 15.68 Positive Gram forms are positive sums of real characters

open in the book · parts/02-mathematical-methods/13-lie-algebra-expansions.tex:2477 · p. 628

Rests on

depends_on definition 15.66 Gram form of a functional ¶
- depends_on theorem 15.54 Completeness in the bilinear case ¶
  - depends_on definition 15.3 Tensor product Lie algebra ¶
  - depends_on lemma A.63 The invariant bilinear forms of an absolutely simple algebra form a line ¶
    - depends_on definition A.62 Absolutely simple ¶
    - proves proof app:A-long-proofs@proof-48 ¶
  - depends_on lemma 15.50 Every $k$-trace comes from a linear functional ¶
    - depends_on definition 15.49 $k$-trace ¶
    - proves proof ch:13-lie-algebra-expansions@proof-29 ¶
  - proves proof ch:13-lie-algebra-expansions@prooflink-1 ¶
depends_on lemma 15.12 Artin decomposition ¶
- depends_on notation 15.1 Standing conventions of this chapter ¶
- depends_on theorem 8.19 Fundamental theorem of algebra ¶
  - depends_on theorem 8.18 Liouville ¶
    - depends_on proposition 8.10 Fundamental theorem for contours ¶
      - depends_on equation 8.6 eq:cpx-contour-integral ¶
      - depends_on theorem 7.43 Fundamental theorem of calculus, II ¶
      - proves proof ch:06-complex-analysis@proof-5 ¶
    - depends_on theorem 8.17 Derivatives of all orders; Cauchy estimates ¶
      - depends_on lemma 8.9 ML estimate ¶
      - depends_on theorem 8.16 Cauchy integral formula ¶
      - proves proof ch:06-complex-analysis@proof-11 ¶
    - proves proof ch:06-complex-analysis@proof-12 ¶
  - proves proof ch:06-complex-analysis@proof-13 ¶
- proves proof ch:13-lie-algebra-expansions@proof-7 ¶
depends_on lemma 15.67 Multiplication is $B_{\psi}$-self-adjoint ¶
- depends_on definition 15.66 Gram form of a functional ¶ ↺
- proves proof ch:13-lie-algebra-expansions@proof-36 ¶
proves proof ch:13-lie-algebra-expansions@proof-37 ¶

Supports

depends_on corollary 15.69 No positive pairing survives the radical ¶

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`	→	Gram form of a functional	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:2495
`depends_on`	→	Artin decomposition	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:2495
`depends_on`	→	Multiplication is $B_{\psi}$-self-adjoint	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:2495
`depends_on`	←	No positive pairing survives the radical	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:2579
`proves`	←	ch:13-lie-algebra-expansions@proof-37	declared	parts/02-mathematical-methods/13-lie-algebra-expansions.tex:2498