Show that it is not a Banach Space

Question

The vector space of all finite sequences $x = (x_n)$ (i.e. $x_n = 0$ for all but finitely many indices $n \epsilon N$) is a normed space with respect to $||x||_{\infty} := sup_{n \epsilon N} |x_n|$.

I haven't really understood Banach spaces well, So I'm really stuck here!

For a good exposition to Banach spaces, see Kolmogorov and Fomin's "Introductory Real Analysis". — user8960, Oct 05 '16 at 21:36

score 1 · Answer 1 · answered Oct 05 '16 at 21:27

1

Consider the sequence defined by $ x_n = (1, 1/2, 1/3, \ldots, 1/n, 0, \ldots) $. Show that $ x_n $ is Cauchy, but has no limit in your space.

answered Oct 05 '16 at 21:27

Ege Erdil

17,747

Show that it is not a Banach Space

1 Answers1