Is an infinite dimensional vector space complete under the norm $\|x\|=\sum|x_i|$

Question

Suppose $\{e_i\}_{i\in I}$ be a Hamel basis of infinite dimensional vector space $X$.

Define the norm $\|x\|=\sum|x_i|$ where $x=\sum x_ie_i$.

Is this normed space complete?

Answer 1

The space is not complete. Without loss of generality, suppose that $\Bbb N \subset I$. It suffices to show that the sequence $$ x^{(n)} = \sum_{i=1}^n \frac{1}{2^i} e_i $$ is Cauchy, but non-convergent.

Answer 2

In general it is not complete. For example consider the space of the definitely null sequences: $$ X=\{x:=(x_n)\;|\;\exists N\in\mathbb N: x_m=0\;\forall m\geq N\}. $$ This is an infinite-dimensional space. In this case, you can take $$ e_1=(1,0,0,0\ldots),\quad e_2=(0,1,0,0\ldots),\quad \ldots $$ Consider the sequence $x^k$ defined by $$ (x^k)_n= \left\{ \begin{array}{ll} 2^{-n},\quad \textrm{if }n\leq k\\ 0 \quad \textrm{if }n> k. \end{array} \right. $$ This is a Cauchy sequence in the norm $\|\cdot\|$, but its limit is not in $X$. In fact, its limit cannot be anything else but the sequence $\bar x$, with $$ (\bar x)_n= 2^{-n} $$ and, of course, $\bar x\not\in X$.