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Suppose $X$ is a reflexive complex Banach space, $S:X\rightarrow \mathbb{C}$ is a Gateaux-differentiable function, and consider the variational problem $$ \text{find } u\in X, \quad\text{such that } \forall v\in X, \quad S'(u,v) = 0, $$ where $S'(u,\cdot)\in X^*$ is the Gateaux differential.

Suppose further we have a sequence of nested subspaces $X_n\subset X_{n+1}\subset \cdots \subset X$ such that every $u\in X$ can be approximated by a sequence $\{u_n\}\in X$, $u_n\in X_n$.

Suppose $x_n$ is the solution of the problem $$ \text{find } u_n\in X_n, \quad\text{such that } \forall v_n\in X_n, \quad S'(u_n,v_n) = 0, $$

We assume that all the $X_n$-problems have a locally unique solution, similarly for the full problem.

My question: what are the minimal conditions on $S$ and/or the spaces $X_n$ and $X$, such that $u_n\rightarrow u$? What mode of convergence can we expect?

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