Show $\ell_\infty (M)$ is a Banach Space

Question

I'm working on problems from Carothers' Real Analysis. The following problem is in the section on completions.

Given any metric space $(M,d)$, check that $\ell_\infty(M)$ is a Banach space.

where $\ell_\infty(M) $ denotes the collection of all bounded, real-valued functions $f: M\to \mathbb{R}$.

Thoughts:

I know that $\ell_\infty$ is a Banach space and that the norm is still the supremum norm $$||f||_\infty=\displaystyle\sup_{x\in (M,\,d)}|f(x)|$$

I'm not too sure where to begin. This is in the section on completions, but I don't see where I would use anything about completions for this problem so a hint would be welcome. Thanks.

Answer 1

You should try to prove directly that the set is a Banach space by showing it is complete. That is, show that for any Cauchy sequence of bounded real-valued functions on $M$, it has a limit that is also a bounded real-valued function on $M$. One big hint I can give you is that you already know that $\mathbb{R}$ is complete.

Answer 2

Hint: Let $\{f_n\}_{n \in \mathbb{N}}$ be a cauchy sequence in $\ell^\infty(M)$

For any $x \in M$ lets define $$ f(x) = \lim_{n \to \infty} f_n(x) $$

Answer 3

Let $\{f_n\} \in (\ell_\infty(M), ||\cdot||_\infty)$ be a Cauchy sequence. Then $\forall x\in M,f_n(x)$ is a Cauchy sequence of $\mathbb{R}$. Since $\mathbb{R}$ is complete, the sequence converges, say to $f(x)$. This defines a $f$ for each $x\in M$. Then you just need to show $\{f_n\} \rightarrow f$ in $||\cdot||_\infty$ (by using the definition $\{f_n\}$ is Cauchy in $||\cdot||_\infty$ and pass one of $m,n$ to $\infty$) and we can then get $f\in \ell_\infty(M)$ by uniform convergence.