This is the last round of questions, I promise.

(1) Identify the space of all continuous linear functionals on the space $\displaystyle c_0$.

(2) $\displaystyle X=(C([0, 2], \mathbb{R}), || . ||_2)$,

$\displaystyle Y=\{f \in C([0, 2], \mathbb{R}: f(1-x)=f(1+x), x \in [0,1]\}$

Is Y closed vector space in X? What is the complement of Y?

(3) Prove that the series

$\displaystyle \sum_{k=1}^{\infty}x_k b_k$ converge for every vector

$\displaystyle x=(x_k) \in l^p, 1<p<\infty$ if and only if $\displaystyle b=(b_k) \in l^q, 1/p +1/q=1$.

These are the last ones I don't know what to do with, and if anyone has the time and is willing to help, I would be really grateful.