Is this space a banach space?

Question

Hi I want to find out whether $l^1$ with the norm $||x||:=sup_n |\sum_{i=1}^{n} x_i|$ is a Banach space. In case that you think that it is a Banach space, just say: It's a Banach space(and then I will first try to prove this), but in case that it is not, I would be grateful to you, if you could give me a sequence, that destroys the Banach space question.

Answer 1

Let $e^i$ be the sequence with a $1$ in the $i$-th place and zeros elsewhere.

Let $x^{k} = \sum\limits_{n=1}^k \dfrac{(-1)^{n-1}}{n}e^n$. Then, for $k < m$

$$\lVert x^k - x^m\rVert = \frac{1}{k+1},$$

since the partial sums of the difference $\sum_{n=k+1}^m \dfrac{(-1)^{n-1}}{n}e^n$ are bounded by the first nonzero partial sum, since the nonzero terms have alternating sign and decreasing magnitude.

So $(x^k)$ is a Cauchy sequence, but its "limit", the alternating harmonic series, is not in $l^1$.