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

==== There are two standard, robust ways — you must prove one of them. ====

===== Show the Picard iterates u(n)→uu^{(n)}\to uu(n)→u in the Banach space C([0,T];X)C([0,T];X)C([0,T];X) (or at least show B(u(n),u(n))→B(u,u)B(u^{(n)},u^{(n)})\to B(u,u)B(u(n),u(n))→B(u,u) in L1([0,T];Y)L^1([0,T];Y)L1([0,T];Y)). Then: =====
# By continuity of BBB, B(u(n),u(n))→B(u,u)B(u^{(n)},u^{(n)})\to B(u,u)B(u(n),u(n))→B(u,u) in L1([0,T];Y)L^1([0,T];Y)L1([0,T];Y).
# By the continuity of the semigroup operator-valued map τ↦eντΔ\tau\mapsto e^{\nu\tau\Delta}τ↦eντΔ and the linearity/continuity of the Bochner integral,

∫0teν(t−s)ΔB(u(n),u(n))(s) ds⟶∫0teν(t−s)ΔB(u,u)(s) ds\int_0^t e^{\nu (t-s)\Delta}B(u^{(n)},u^{(n)})(s)\,ds
\longrightarrow
\int_0^t e^{\nu (t-s)\Delta}B(u,u)(s)\,ds∫0teν(t−s)ΔB(u(n),u(n))(s)ds⟶∫0teν(t−s)ΔB(u,u)(s)ds
in the target Banach space. So we can pass the limit inside the integral legitimately.

This is the textbook strategy: obtain strong convergence in the appropriate topology; then interchange limit and integral because the Bochner integral is continuous.

===== If you insist on using some extended-limit operator LLL instead of actual norm limits, you must show: =====
* LLL extends to a bounded linear functional on L1([0,T];X)L^1([0,T];X)L1([0,T];X) (or on XXX after integration); or
* LLL is continuous with respect to the topology in which your integrals converge.

Only then can you move LLL across the integral by linearity/continuity. Without such a continuity property, LLL may fail dramatically (it can map convergent sequences to divergent ones or vice versa).