Given $p \in [1, + \infty]$, let $c = (c_n)$ a sequence of reals such that $\forall x = (x_n) \in l^{p}$ the series $ \phi(x) = \displaystyle\sum_{n=1}^{\infty} c_n x_n$ converges. Show that $\phi \in (l^p)'$ (Which means $\phi$ is a linear continuous functional).
I already proved that $l^{p}$ spaces for $1 \leq p < \infty$ are separable, maybe I could use this, but can't see how. Perhaps I can use Banach-Steinhaus theorem in this problem, but also don't know how.
Any help would be appreciated.
Thanks.
The key steps: Define the family of linear functionals $F =\lbrace f_n | n \in \mathbb{N} \rbrace$ such that $f_n x=\sum_{i=1}^n c_i x_i$ for all $x \in l^p$. This enables you, for each n, to bound $|f_n x|$ by a constant multiple of $||x||_p$ using Hölder's inequality for finite sums. We need this finite, as a q-Norm of $(c_n)_{n \in \mathbb N}$ is not given to exist. Now you know that everyone in $F$ is bounded. Look at the behaviour of functionals from $F$ at every $x \in X$ to establish the second condition for the Banach-Steinhaus Theorem. Obtain the uniform bound and use it to establish close-enough approximation of $\phi$ by some $f \in F$ to conclude with a bound for $||\phi||_{l^{p'}}$.