Theorems · Theorem · functional analysis

IsMaxFilter.norm_add_sameRay

∀ {α : Type u_1} {E : Type u_3} [inst : SeminormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] {f : α → E} {l : Filter α}
  {c : α} {y : E}, IsMaxFilter (norm ∘ f) l c → SameRay ℝ (f c) y → IsMaxFilter (fun x => ‖f x + y‖) l c

If f : α → E is a function such that norm ∘ f has a maximum along a filter l at a point c and y is a vector on the same ray as f c, then the function fun x => ‖f x + y‖ has a maximum along l at c.

Defined in

Mathlib.Analysis.Normed.Module.Extr

Cited by

4 results in Mathlib

Foundations

Depth 159 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SeminormedAddCommGroupNormedSpace