“How to measure the distance between two functions?”

Let $V = \{f: f: [0,1] \to \R\}$ be the space of bounded functions from $[0,1]$ to $\R$, we will consider various ways to measure size of a function

• sup norm: $ \| f \|_\infty = \sup \{ f(x) : x \in [0,1]\} $
• $L^1$ norm $ \| f\|_1 = \int |f(x)| dx $
• $L^2$ norm $ \| f\|_2 = (\int |f(x)|^2 dx)^{1/2}$
• in general, for $1 \leq p < \infty$, we have $L^p$ norm.

The different norms equip the vector space $V$ with different topologies. This is different from the finite dimensional vector space case.

In sport, sometimes you measure the score using the best score (like freestyle skiing in the winter Olympics); sometimes you use the average score of several try (like Tour de France, you add up the points in each leg); this is like sup or $L^1$ norm.

In this class, we will consider sup norm, and we define distance as $$ d_\infty(f,g) = \| f-g\|_\infty $$

We say a sequence of functions $f_n$ converge to $f$ uniformly, if $\lim_n d_\infty(f_\infty, f) = 0$.

Q: can you find a metric on the space of functions $V$ so that metric convergence means pointwise convergence? (No, you but you can still define a topology on $V$ so that convergence in that topology means pointwise convergence)