Determining if metric space is complete

Question

Is this subset a complete metric space with the $\rho_{\infty}$ metric?

$\{f \in C[a,b] | f(a)=0\}$

I know that I must show that a Cauchy sequence has a limit in the same metric space, but I don't know how to do this. Also, what is the significance of metric choice here?

Answer 1

You need to use

1. That uniform convergence preserves continuity, i.e. that if a sequence of continuous function $f_n$ converges pointwise and uniformly to $f$, then $f$ is continuous.

2. That convergence in the supremem metric $\rho_\infty$ means pointwise and uniform convergence.

The significance of the metric is that this only works for the supremem metric. In the metric $$ \rho_2(f,g) = \int_0^2 (f(x) - g(x))^2 \,dx \text{,} $$ for example, you have that $$ \frac{x^n}{1 + x^n} \to \begin{cases} 1 &\text{if $x > 1$} \\ \tfrac{1}{2} &\text{if $x=1$} \\0 &\text{if $x < 1$} \end{cases} \quad \text{ as $n \to \infty$} $$

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You start your proof with a cauchy sequence $f_n$ in your metric space.

First, you argue that $f_n(x)$ is then a cauchy sequence in $\mathbb{R}$, for every $x$. This works because the supremum metric by definition says that if $d(f,g) \leq \epsilon$, then $|f(x) - g(y)| < \epsilon$ for all $x$. Since we know that $\mathbb{R}$ is complete, this proves that $f_n$ converges pointwise to some function $f$. This also shows that if $f_n(a) = 0$ for all $n$, then $f(a) = 0$ as well.

Then, we use the definition of the supremem metric again to show that this pointwise limit is uniform in $x$. Again, we use that $d(f,g) \leq \epsilon$ means $|f(x) - g(y)| < \epsilon$ for all $x$.

We thus know that $f$ is continuous, since $f$ is the uniform limit of continuous functions, and that completes the proof.

Answer 2

Hint: if a sequence of continuous functions converges is the sense $\|\cdot\|_\infty$ ($\rho_\infty$ in your notations), then it converges pointwise.

Further details.

1) $C[a,b]$ with metric $\rho_\infty$ is a complete space. This is covered in all books.

2) If a sequence of functions converges uniformly (i.e. in the metric $\rho_\infty$), then it converges pointwise. Follows from the definition of this metric.

3) Suppose we have a Cauchy sequence $\{f_n\}$ of functions in $L\subset C[a,b] $. Then $f_n$ converges in $C[a,b]$ by (1). By (2) we obtain pointwise convergence of $f_n$, in particular, the sequence $f_n(a)$ converges. Clearly, this sequence is identically zero, therefore the limiting function $f$ also belongs to $L$.