Theorems · Theorem · functional analysis

Module.Basis.exists_opNNNorm_le

∀ {𝕜 : Type u} [inst : NontriviallyNormedField 𝕜] {E : Type v} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] {F : Type w} [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F] [CompleteSpace 𝕜]
  {ι : Type u_1} [Finite ι] (v : Module.Basis ι 𝕜 E),
  ∃ C > 0, ∀ {u : E →L[𝕜] F} (M : NNReal), (∀ (i : ι), ‖u (v i)‖₊ ≤ M) → ‖u‖₊ ≤ C * M

A weaker version of Basis.opNNNorm_le that abstracts away the value of C.

Defined in

Mathlib.Analysis.Normed.Module.FiniteDimension

Cited by

1 results in Mathlib

Foundations

Depth 174 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceCompleteSpaceFinite