Limit of Fréchet differentiable functions

Question

If I have a sequence of Fréchet differentiable functions, under what conditions can I prove that the limit is Fréchet differentiable? For example, suppose that I have $f:\ell^p(\mathbb{R})\to\mathbb{R}$ defined by an expression $f(x)=\sum_{k=0}^\infty f_k(x_k)$ when $f_k:\mathbb{R}\to\mathbb{R}$ is Fréchet differentiable. Then how I can prove that $Df(x)(d)=\sum_{k=0}^\infty Df_k(x_k)(d_k)$?

Answer 1

Suppose we have the following (strong) condition: Suppose there is some $B \in l_q$ and for any $\epsilon> 0$ there is some $\delta>0$ such that for any $k$ and for any $|d| < \delta$ we have $|f_k(x_k+d)-f_x(x_k)-f_k(x_k)d| \le \epsilon |B_k||d|$.

Choose $x \in l_p, d \in l_q$ and $\epsilon > 0$. Suppose $\|d\|_q < \delta$ (in particular, $|d_k| < \delta$).

Then \begin{eqnarray} |f(x+d)-f(x)-\sum_k f_k'(x_k)d_k | &\le& \sum_k |f_k(x_k+d_k)-f_k(x_k)-f_k'(x_k)d_k | \\ &\le & \sum_k \epsilon |B_k||d_k| \\ &=& \epsilon \|B\|_p \|d\|_q \end{eqnarray} Then $f$ is differentiable and $Df(x)d = \sum_k f_k'(x_k)d_k$.