English
Let f be a sequence and l its Abel limit; then for suitable parameters the Abel sums converge to l along Stolz sets and cones in appropriate spaces E, F.
Русский
Пусть f — последовательность, и ее предел Абеля в обобщенном виде достигается; тогда сумма Абеля сходится к l вдоль множеств Stolz в нужном пространстве.
LaTeX
$$$$\\forall f, l, \\text{(appropriate hypotheses)}, \\operatorname{Tendsto} \\left( z \\mapsto \\sum_{n=0}^{\\infty} f(n) z^{n} \\right) (\\mathcal{N}(l)).$$$$
Lean4
/-- **Maximum modulus principle** on a connected set. Let `U` be a (pre)connected open set in a
complex normed space. Let `f : E → F` be a function that is complex differentiable on `U` and is
continuous on its closure. Suppose that `‖f x‖` takes its maximum value on `U` at `c ∈ U`. Then
`‖f x‖ = ‖f c‖` for all `x ∈ closure U`. -/
theorem norm_eqOn_closure_of_isPreconnected_of_isMaxOn {f : E → F} {U : Set E} {c : E} (hc : IsPreconnected U)
(ho : IsOpen U) (hd : DiffContOnCl ℂ f U) (hcU : c ∈ U) (hm : IsMaxOn (norm ∘ f) U c) :
EqOn (norm ∘ f) (const E ‖f c‖) (closure U) :=
(norm_eqOn_of_isPreconnected_of_isMaxOn hc ho hd.differentiableOn hcU hm).of_subset_closure hd.continuousOn.norm
continuousOn_const subset_closure Subset.rfl
