Show that closed balls is not bounded in the space of all bounded functions.
Let X be infinte. And let B(X) be the set of all bounded functions f:X$\rightarrow\mathbb{R}$ on X.
and equip it with the metric
d(f,g)=sup $\left \{ |f(x)-g(x)| : x\in X \right \}$
Show that for r>0 and $f\in$ B(X) the closed ball $B_{r}(f)$ is not compact.
I think I have to find a sequence of functions such that its limit is not bounded . If I consider $f_n:(0,1)\rightarrow\mathbb{R}$ such that $f_n$(x)= $\frac{n}{nx+1}$ then
|$f_n$|$\leq$n and the point wise limit f(x)=1/x is not bounded in (0,1) but can't put it into a closed ball. Please help me to find this closed ball or if there is any other example you can also provide that. Thank you.
Let $\{x_n\}_{n\in\Bbb{N}}$ be a countably infinite subset of $X$ and for $n \in \Bbb{N}$ let $f_n : X \to \Bbb{R}$ be a function such that $$f(x_n) = r, \quad f(x) = 0, \forall x \in X \setminus \{x_n\}.$$ Then $(f_n)_n$ is a sequence in the closed ball $B_r(X)$ which has no convergent subsequence since $d(f_m,f_n) = r$ for all $m \ne n$.