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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.

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    No problem, and welcome to Math Stackexchange.2019-01-13

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No, $V$ it self is too small. Look at $f_n(x)=\max(1,1/(n\cdot x)),\qquad \qquad n=1,2,\ldots$ it converges pointwise, but the limit is not in $V$.

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    @Bana: Show that it is not [first countable](http://en.wikipedia.org/wiki/First-countable_space).2011-10-19