Theorems · Theorem · commutative algebra

seminormFromBounded_bddAbove_range

∀ {R : Type u_1} [inst : CommRing R] {f : R → ℝ} {c : ℝ},
  0 ≤ f → (∀ (x y : R), f (x * y) ≤ c * f x * f y) → ∀ (x : R), BddAbove (Set.range fun y => f (x * y) / f y)

If f : R → ℝ is a nonnegative, multiplicatively bounded function, then for every x : R, the image of y ↦ f (x * y) / f y is bounded above.

Defined in

Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded

Cited by

5 results in Mathlib

Foundations

Depth 111 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRing