toSeminorm_comp — Mathlib

English

If s = {a}, the polar is precisely those functionals with ‖· a‖ ≤ 1.

Русский

Если s = {a}, поляр равен тем функционалам, для которых ‖x a‖ ≤ 1.

LaTeX

$$$\operatorname{polar}_{\mathbb{K}}(\{a\}) = \{ x \in \mathrm{StrongDual}_{\mathbb{K}}(E) \\| \|x(a)\| \le 1 \}$$$

Lean4

theorem toSeminorm_comp (f : F →ₗ[𝕜] 𝕜) (g : E →ₗ[𝕜] F) : f.toSeminorm.comp g = (f.comp g).toSeminorm :=
  by
  ext
  simp only [Seminorm.comp_apply, toSeminorm_apply, coe_comp, Function.comp_apply]

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