Metric on function space for Pointwise Convergence We say that $(f_n)_{n\in\Bbb N}$ is a sequence of functions from $[a,b]$ to $\mathbb{R}$ that converges pointwise to $f\colon[a,b]\longrightarrow\mathbb{R}$ if, for any $x$ in $[a,b]$, we have $\lim_{n\to\infty}f_n(x)=f(x)$.
In this definition we talked about “convergence” without speaking of metric. Is there any metric (or topology) on space of functions (or continuous functions) from $[a,b]$ into $\mathbb{R}$ which leads to this convergence?
 A: There is such a topology, yes. For each $x\in[0,1]$, let $\Bbb R_x=\Bbb R$ and consider the product $\prod_{x\in[0,1]}\Bbb R_x$, endowed with the product topology. Then $(f_n)_{n\in\Bbb N}$ converges to $f$ with respect to this topology if and only if it converges pointwise.
Now, I am going to provide an outline of a proof due to M. K. Fort that no metric can be defined in your space with that property. For more details, see his article A note on pointwise convergence. Without loss of generality, we can assume that $[a,b]=[0,1]$. Fort defines a family $\{f_{n,m}\mid n,m\in\Bbb N\}$ of continuous functions from $[0,1]$ into $\Bbb R$. This family is defined in such a way that, for each $n\in\Bbb N$ and each $t\in[0,1]$, $f_{n,m}(t)=0$ for all but three values of $m$. Therefore, for each $t\in[0,1]$ we have $\lim_{m\to\infty}f_{n,m}(t)=0$. In other words, for each $n\in\Bbb N$, the sequence $(f_{n,m})_{m\in\Bbb N}$ converges pointwise to the null function. So, if there was a metric $d$ on $\Bbb R^{[0,1]}$ such that convergence with respect to $d$ would be the same thing as pointwise convergence, we would have $(\forall n\in\Bbb N):\lim_{m\to\infty}f_{n,m}=0$ in $\bigl(\Bbb R^{[0,1]},d\bigr)$. Therefore, it is possible to choose integers $N_n$ such that$$m_n>N_n\implies d\left(f_{n,m_n},0\right)<\frac1n.$$So, any sequence $\left(f_{n,m_n}\right)_{n\in\Bbb N}$ converges pointwise to the null function.
Then Fort shows that this is not possible. It follows from the way that the functions $f_{n,m}$ are defined that, for each $n\in\Bbb N$, there is some closed interval $[a_n,b_n]$ on $[0,1]$ such that $f_{n,m}|_{[a_n,b_n]}=1$ if $m$ is larger than some natural number $m_n$ and that $[a_{n+1},b_{n+1}]\subset[a_n,b_n]$. But then $\bigcap_{n\in\Bbb N}[a_n,b_n]\ne\emptyset$ and, if $x\in\bigcap_{n\in\Bbb N}[a_n,b_n]$,$$\lim_{n\to\infty}f_{n,m_n}(x)=1\ne0.$$So, the sequence $\left(f_{n,m_n}\right)_{n\in\Bbb N}$ does not converge pointwise to the null function.
