How do I exhibit an open cover of the closed unit ball of the following:
(a) $X = \ell^2$
(b) $X=C[0,1]$
(c) $X= L^2[0,1]$
that has no finite subcover?
How do I exhibit an open cover of the closed unit ball of the following:
(a) $X = \ell^2$
(b) $X=C[0,1]$
(c) $X= L^2[0,1]$
that has no finite subcover?
Suppose $\{y_n\}_{n=1}^\infty$ is a sequence in the closed unit ball $B$ which has no accumulation point. Then $U_{N} = B \smallsetminus \{y_n\,:\,n \geq N\}$ is open and $B \subset \bigcup_{N=1}^\infty U_N$. However, $\{U_{N}\}_{N=1}^\infty$ has no finite subcover.
For a) take the standard basis, for b) take $y_{n}(t) = t^n$ and for c) try $y_n(t) = \exp{(2\pi i \,n \,t)}$.