Editing Openai/6935580c-34e4-8001-9136-7bbdf1499790 (section)

==== If your space is the nonnormed extended algebra EEE and you want to assert the mild expression exists there, you must add precise axioms for EEE. At minimum you need: ====
* A topology (or net/sequence notion) and a Banach-type seminorm ∥⋅∥X\|\cdot\|_X∥⋅∥X on the subspace where the Picard iterates live, or an explicit statement that the semigroup operators eνtΔe^{\nu t\Delta}eνtΔ act and satisfy a bound of the form in (H1).
* Closure of EEE under multiplication with a bilinear estimate like (H2).
* The ability to form Bochner (or primitive-based) integrals in EEE and that your linear limit LLL commutes with these integrals.

Without explicit axioms of this type you cannot rigorously show existence of the Bochner integral in EEE; the claim “the mild equation is well-defined in the extended space” is therefore conditional on supplying those axioms.