Problem with a sequence of continuous functions in $C^0([0,1])$ with integral 1-norm

Question

Consider the space $C^0([0,1])$ together with the norm $$||f|| = \int_{0}^{1} | f(t) | dt$$ Take the sequence in $C^0([0,1])$ defined by $$u_n(t) = t^n$$

During a lecture my professor said that it's simple to show that this is a Cauchy sequence (which I agree) and then he added that the sequence does not converge to an element of $C^0([0,1])$. In particular he said that $u_n$ converges to $u$, where $u(t) =0$ if $t \in [0,1)$ and $u(t) =1$ if $t=1$. How can I prove that this function is really the limit of $u_n$?

Answer 1

$\int_0^{1} |u_n(t)-u(t)| dt =\frac 1 {n+1} \to 0$. So $u_n \to u$ in $L^{1}$. However the sequence is actually convergent in $C^{0}[0,1]$ to the zero function.

If you want an example of a Cauchy sequence which is not convergent take $u_n(t)=0$ for $t \leq \frac 1 2$, $u_n(t)=1$ for $t \geq \frac 1 2 +\frac 1 n$and $u_n(t)=n(t-\frac 1 2)$ for $\frac 1 2 \leq t \leq \frac 1 2+\frac 1 n$

Answer 2

There are multiple notions of convergence, and different notions give different answers on whether this sequence converges, and what it converges to.

One notion is pointwise convergence. That's when we just look at an arbitrary input value $t$ in the domain of out functions, and look at the sequence $u_1(t), u_2(t), \ldots$ and see if they converge to anything. If they do, then that's what we set $u(t)$ to. In your case, the $u_n$ do converge to the $u$ given, and that $u$ is not $C^0$, so the sequence does not converge pointwise in $C^0[0,1]$.

If a sequence of functions with real values converges pointwise, then that limit is unique. In general, any codomain where convergent sequences of points have unique limits will yield unique pointwise limits of functions. In particular any metric space.

On the other hand, we are also given a norm. And once we have a norm (or any other notion of distance between functions), we can also ask whether functions converge in this norm. And norms might not see differences between functions that pointwise study does. In particular, under this norm, your sequence $u_n$ converges to the zero function, because the distance between $u_n$ and the zero function goes to $0$ as $n$ increases. So by the norm, this sequence is actually convergent in $C^0[0,1]$.

And in fact, this limit is also unique. The only continuous function that $u_n$ converges to, according to the norm, is the zero function.