Let $X=C[0,1]$ and $W=\{f\in X\mid f(0)=0\}$. What is the closure of $W$ wrt the 1-norm $\|\cdot\|_1$.
My solution is as follows:
The closure is the whole space $X$. To see this take any function $f(x)\in X$, assume with out loss of generality that $f$ is in the upper right plane. We will construct a sequence that converges to this $f(x)$.
Consider the sequence $f_\epsilon(x)= \begin{cases} f(x), & \text{if }x\in [0,1]-(0,\epsilon), \\ \frac{f(\epsilon)}{\epsilon}x, & \text{if }x\in(0,\epsilon). \end{cases}$
Then if we let $\epsilon\rightarrow 0$ $f_{\epsilon}\rightarrow f$ and so we have that the limit points are the whole of $X$.
Is this correct?
Thanks for any help