I am currently facing the following problem in a field where I lack expertise, maybe even to the point of being oblivious to the basics. I would be grateful for any hints and in particular for references.
Setting: Consider reflexive Banach spaces $B$ and $\tilde{B}$ and a smooth non-linear map $\Xi: B \to \tilde{B}$. Let $$M = \Xi^{-1}\left( \{ 0 \} \right).$$ From results found in Lang's book on differential geometry, we can deduce that $M$ is a Banach-submanifold if for every $x\in M$ the linear map $D\Xi(x)$ is surjective and its kernel is complemented in $B$. By using the smooth charts of $M$, one gets the following continuity property for the tangent spaces:
For every sequence $\{ x_n \}_{n \in \mathbb{N}} \subset M$ that converges to some $x \in M$ and every $v \in Tan_x M$ there exists a sequence $\{ v_n \}_{n \in \mathbb{N}}$ with $v_n \in Tan_{x_n}M$ that converges to $v$.
Question: I would like to know under what conditions on $\Xi$ this also holds for (norm-)convergence replaced with weak convergence. The statement I'd like to show then reads:
For every sequence $\{ x_n \}_{n \in \mathbb{N}} \subset M$ that converges weakly to some $x \in M$ and every $v \in Tan_x M$ there exists a sequence $\{ v_n \}_{n \in \mathbb{N}}$ with $v_n \in Tan_{x_n}M$ that converges weakly to $v$.
For example, if that helps, we could also assume that $\Xi$ is smooth as a map $B_{\nu} \to \tilde{B}_{\nu}$ for Banach spaces that are compactly embedded into $B$ and $\tilde{B}$, respectively, and also for Banach spaces $B_{\mu}$, $\tilde{B}_{\mu}$ into whom $B$ and $\tilde{B}$ embed compactly. Think for example of fractional Sobolev spaces $W^{s,p}$ for a range of parameters $s$.