Let $X$ be a vector space (over $\mathbb{R}$) consisting of all real-valued sequences, which have almost all elements equal to $0$.
For $x:= (x_n)_{n\in \mathbb{N}}^{\infty} \in X$ we define its norm as $N(x):= \sup{ \left \{ \frac{\left | x_n \right |}{n}: n\in \mathbb{N} \right \}}$.
Is $(X, N)$ a Banach space?
A sequence $(y_n)_{n\in \mathbb{N}}^{\infty} \subset X$ is a Cauchy sequence, if for all $\varepsilon >0$ and $k, l$ large enough, $N(y_k – y_l) < \varepsilon$, but here I have problems with understanding what is a norm of such difference since I need to consider a sequence of sequences. I think this may not be Banach since the norm is defined with supremum of nonzero elements and it could possibly tend to something different than zero, but I am confused with double indices.
Best Answer
No the sequence $u_n$ such that $u_n(m)=1, m\leq n$ and $u_n(m)=0, m>n$ is a Cauch sequence which does not converge.