# A Continuous function $f: \overline{B_1(0)} \subset \ell^2\to \mathbb{R}$ which does not reach the maximum?

If necessary, recall that $$\ell^2 = \{x=\{x_n\}_n\subset \mathbb{R} : \|x\|^2:=\sum_n |x|^2<\infty\}$$

and $$\overline{B_1(0)}$$ is the closed unit ball with respect to that norm.

Can we find an explicit example of a continuous function $$f: \overline{B_1(0)} \subset \ell^2\to \mathbb{R}$$ which does not attain its maximum?

The point is that this ball is not a compact set.

Thank you.

• As per my previous comment, your question still doesn't make the most sense. Can you maybe rephrase the exact question of the exercise in the body ? – Rebellos Mar 1 at 18:50
• No, sorry, I did a mistake in the question and now is correct. The point is to show a function satisfying the conditions mentioned in the question in order to convince by an example that it is not compact. – Rubén Fernández Fuertes Mar 1 at 18:50
• So it comes down to showing that $\overline{B_1^{\ell_2}}$ is not compact ? – Rebellos Mar 1 at 18:51
• I said no, and I say, again, no. I know by another proof that it is not compact. I repeat, the point is to give an example of this function. – Rubén Fernández Fuertes Mar 1 at 18:52
• Note that $f : \overline{B_1(0)} \in \ell_2$ isn't a solid expression for a function and this is why I am struggling to understand. Do you mean that you want to construct a function with its domain being the Unit Ball of $\ell_2$ such that it is a sufficient example in showing that it cannot be compact ? That's pretty straightforward using the defintiion of a compact operator. – Rebellos Mar 1 at 18:54

Try $$f(x) = \sum_{n=1}^\infty (1-1/n) x_n^2$$ Note that $$f(x) < 1$$ for all $$x \in \overline{B_1(0)}$$, and you can get arbitrarily close to $$1$$...