English
The map toLp is uniform inducing, given the Lipschitz and uniform continuity properties.
Русский
Карта toLp является равномерно индуцирующей, при условии липшица и равномерной непрерывности.
LaTeX
$$IsUniformInducing (@toLp p (Π i, β i))$$
Lean4
/-- seminormed group instance on the product of finitely many normed groups, using the `L^p`
norm. -/
instance seminormedAddCommGroup [∀ i, SeminormedAddCommGroup (β i)] : SeminormedAddCommGroup (PiLp p β) :=
{ Pi.addCommGroup with
dist_eq := fun x y => by
rcases p.dichotomy with (rfl | h)
· simp only [dist_eq_iSup, norm_eq_ciSup, dist_eq_norm, sub_apply]
· have : p ≠ ∞ := by
intro hp
rw [hp, ENNReal.toReal_top] at h
linarith
simp only [dist_eq_sum (zero_lt_one.trans_le h), norm_eq_sum (zero_lt_one.trans_le h), dist_eq_norm,
sub_apply] }
