Let $E$ be a Banach space and $(x_n)_n$ a sequence converging to zero. If I arbitrarily decompose $E$ as $E=E_1\bigoplus E_2$ does the sequence $(x^1_n)_n$ necessarily tend to zero in $E_1$ ? I have no assumption on the dimensions of $E_1$ and $E_2$.
I know that the answer is no if the direct sum involves and infinite amount of terms. Here, there is a finite number of terms but their dimensions may well be infinite.
I think the answer is also no, but this leads me to a problem.
What I am really doing is trying to prove that if a function $f:E \rightarrow F$ has continuous partial (w.r.t. $E_1$ and $E_2$) differentials then it is differentiable. So I expand as follows : $f(a+h)=f(a_1 + h_1, a_2 + h_2)=f(a_1+h_1,a_2)+D_2f(a_1+h_1,a_2)(h_2)+o_{h_2 \rightarrow 0}(h_2)$ and I have read a proof (not online unfortunately) which seems to imply the use of something like converting $o_{h_2 \rightarrow 0}(h_2)$ to $o_{h \rightarrow 0}(h)$. Which I have my doubts about. If the answer to my question abive is yes, then there is a problem, right ?
Then again, in my question, I have a normed space and nothing is known about the restriction of that norm on the subspaces in the decomposition. In calculus, I presume that the subspaces themselves have a norm and that implicitly the finite direct sum is normed with a canonical norm (e.g. a $p$-norm or at least one that equips $E$ with the product topology). Is there such a convention implied in textbook calculus theorems ? (A "yes" to this question is just as good to me as an answer to the actual question).