English
For p with 0 < p.toReal and x ∈ WithLp p (α × β), the norm decomposes as a p-norm of the components: ||x|| = (||idemFst x||^{p.toReal} + ||idemSnd x||^{p.toReal})^{1/p.toReal}.
Русский
Для p с 0 < p.toReal и x ∈ WithLp p (α × β) норма разлагается как p-норма компонент: ||x|| = (||idemFst x||^{p.toReal} + ||idemSnd x||^{p.toReal})^{1/p.toReal}.
LaTeX
$$$\|x\| = \bigl( \|\mathrm{idemFst}\,x\|^{p.toReal} + \|\mathrm{idemSnd}\,x\|^{p.toReal} \bigr)^{1/p.toReal}.$$$
Lean4
/-- The canonical map `WithLp.equiv` between `WithLp ∞ (α × β)` and `α × β` as a linear isometric
equivalence. -/
def prodEquivₗᵢ : WithLp ∞ (α × β) ≃ₗᵢ[𝕜] α × β
where
__ := WithLp.equiv p _
map_add' _f _g := rfl
map_smul' _c _f := rfl
norm_map' := prod_norm_toLp
