Theorems · Theorem · functional analysis

Seminorm.continuous_of_forall

∀ {𝕝 : Type u_6} {E : Type u_7} [inst : SeminormedRing 𝕝] [inst_1 : AddCommGroup E] [inst_2 : Module 𝕝 E]
  [inst_3 : TopologicalSpace E] [IsTopologicalAddGroup E] {p : Seminorm 𝕝 E},
  (∀ r > 0, p.ball 0 r ∈ nhds 0) → Continuous ⇑p

A seminorm is continuous if p.ball 0 r ∈ 𝓝 0 for all r > 0. Over a NontriviallyNormedField it is actually enough to check that this is true for some r, see Seminorm.continuous.

Defined in

Mathlib.Analysis.Seminorm

Cited by

1 results in Mathlib

Foundations

Depth 161 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SeminormedRingAddCommGroupModuleTopologicalSpaceIsTopologicalAddGroup