Does $\{e_n\}$ have a cluster point?

Question

Let $X$ be an infinite dimensional Banach space and $\{e_n\}$ is a sequence of basis elements. Is it justified to say $\{e_n\}$ has no cluster point?

I think it is true (looking at euclidean spaces) but I cannot figure out why! Would somebody help me?

Answer 1

Not necessarily.

Take for example $X=\{\,f\in L^2[-\pi,\pi]: f(-x)=-f(x)\}$. Then $$ f_1=\sin x, \, f_n=\sin x+\frac{\sin nx}{n}, \,n>1, $$ is a sequence a basis elements with cluster point $f_1$.