Theorems · Theorem · functional analysis

PiLp.nnnorm_toLp_const

∀ {p : ENNReal} {ι : Type u_2} [hp : Fact (1 ≤ p)] [inst : Fintype ι] {β : Type u_5}
  [inst_1 : SeminormedAddCommGroup β],
  p ≠ ⊤ → ∀ (b : β), ‖WithLp.toLp p (Function.const ι b)‖₊ = ↑(Fintype.card ι) ^ (1 / p).toReal * ‖b‖₊

When p = ∞, this lemma does not hold without the additional assumption Nonempty ι because the left-hand side simplifies to 0, while the right-hand side simplifies to ‖b‖₊. See PiLp.nnnorm_equiv_symm_const' for a version which exchanges the hypothesis p ≠ ∞ for Nonempty ι.

Defined in

Mathlib.Analysis.Normed.Lp.PiLp

Cited by

4 results in Mathlib

Foundations

Depth 222 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FactFintypeSeminormedAddCommGroup