Finding a good example of an element of a given dual space

Question

Let $$\{X=c_0=\{x=(x_j):\lim_{j\longrightarrow\infty}x_j=0\}$$ and the norm be $$||x||_\infty=\underset{1\le j\le\infty}{max}|x_j|.$$

I want to find two examples of elements of the dual space $X^*$.

The first is :

$$s_n=\frac{n+1}{(n+2)^2}$$, since $$\lim_{n\longrightarrow\infty}\sum_{n=0}^\infty s_n =0. $$

Then I form the second by linearity of the dual space elements by adding to $s_n$ another converging sequence with same properties: $$r_n=\bigg(-\frac{1}{2}\bigg)^n$$

where $$\lim_{n\longrightarrow\infty}\sum_{n=0}^\infty r_n=0,$$

So I am then "claiming" that $$(s_n+r_n)=t_n\in X^*.$$

But this can not be correct, since $$\lim_{n\longrightarrow\infty} t_n \ne 0$$

What was wrong with the choice of elements here, and what is a better example?

Answer 1

There are a few mistakes here:

1. You have some weird notation going on. It doesn't make sense to write $\lim_{n \to \infty} \sum_{n=0}^\infty s_n$ because $n$ shouldn't be both the dummy variable in the summation and also the limit variable. I'm guessing you mean to say $\lim_{N \to \infty} \sum_{n=N}^\infty |s_n| = 0$, i.e. $(s_n) \in \ell_1(\mathbb{R})$, since that's a sufficient condition for $(s_n)$ defining an element of $X^*$ via $\phi_{(s_n)} : (x_n) \mapsto \sum_{n=0}^\infty s_n x_n$. I'm not confident enough about your meaning to just edit your post though; it would be nice if you could edit to make the question clearer yourself.

2. Actually, your claim about $s_n$ is not true. In fact, $\sum_{n=0}^\infty s_n$ doesn't even converge. You can check it yourself, but basically it's because $s_n \approx \frac 1 n$ for $n$ large.
3. I don't agree with your claim $\lim_{n \to \infty} t_n \not= 0$. Actually, we have $\lim_{n \to \infty} t_n = \lim_{n \to \infty} (s_n + r_n) = \lim_{n \to \infty} s_n + \lim_{n \to \infty} r_n = 0+0 = 0$. In fact, we can go further: if you fix mistake (2) above by replacing $s_n$ with some other sequence that actually converges, say $s_n = \frac{n+1}{(n+2)^3}$, then you really will end up with $\lim_{N \to \infty} \sum_{n=N}^\infty t_n = 0$. In other words, $t_n$ will be an element of $X^*$.