Possible Duplicate:
Norm for pointwise convergence
Let $V=C([0,1],\mathbf{R})$ be the vector space of continuous real-valued functions on $[0,1]$.
Let $(f_n)$ be a sequence in $V$. Then $(f_n)$ converges with respect to the max-norm on $V$ if and only if $(f_n)$ is uniformly convergent.
As a consequence, since the limit of a uniformly convergent sequence in continuous, we conclude that $V$ endowed with the max-norm is a Banach space.
Now, on $V$ there is also a notion of pointwise convergence. Is there a norm $\Vert \cdot \Vert_{pc}$ on $V$ such that a sequence $(f_n)$ in $V$ converges with respect to $\Vert \cdot\Vert_{pc}$ if and only if $(f_n)$ is pointwise convergent?
Note that $V$ will not be Banach under this norm.