English
If f and g are continuous maps X → L(E,G) and X → L(F,G), then x ↦ f(x) coprod g(x) is continuous on X.
Русский
Если f,g: X → L(E,G) и X → L(F,G) непрерывны, то x ↦ f(x) coprod g(x) непрерывно на X.
LaTeX
$$$f: X \to L(E,G),\; g: X \to L(F,G)\ \text{ и } f,g\text{ непрерывны; } x \mapsto f(x) \coprod g(x)\text{ непрерывно.}$$$
Lean4
/-- A compact operator between two topological vector spaces. This definition is usually
given as "there exists a neighborhood of zero whose image is contained in a compact set",
but we choose a definition which involves fewer existential quantifiers and replaces images
with preimages.
We prove the equivalence in `isCompactOperator_iff_exists_mem_nhds_image_subset_compact`. -/
def IsCompactOperator {M₁ M₂ : Type*} [Zero M₁] [TopologicalSpace M₁] [TopologicalSpace M₂] (f : M₁ → M₂) : Prop :=
∃ K, IsCompact K ∧ f ⁻¹' K ∈ (𝓝 0 : Filter M₁)
