I am working on following task:
Let $$ \ell^1(\mathbb{R}) = \{ x=(x_k)_{k=1}^\infty : x_k \in \mathbb{R}, \|x\|=\sum_{k=1}^\infty |x_k| < \infty \}.$$ Fix $y \in \ell^1$ and define $$ B=\{x \in \ell^1: \|x\|\leq1\}$$ and $$ M = \{x\in \ell^1: |x_k|\leq |y_k|\}.$$
Find a continuous, real valued function on $\ell^1$ that does not have a largest value on $B$ and show that $M$ is a compact subset of $\ell^1$.
My first thought was that such a function doesn't exist, since $B$ is compact and a continous function will take its minimum and maximum value. But then I recognized that that $\ell^1$ is an infinite dimensional space, so the closed unit ball doesn't have to be compact anymore. Do you have some good idea to define such a function?
For the compactness of $M$ I thought about using the definition, that every sequence in $M$ has a convergent subsequence in $M$. If I take a sequence $x^n$ in $M$, then $|x^n_k| \leq |y_k|$ for every $k$ (coordinates) and $n$ (member of the sequence). But how can I construct a convergent subsequence out of $x^n$?
Some hints or solutions would be really helpful.