Check continuity and find norm of linear functional

Question

I have to check continuity and find norm of the following linear functional $$\ell^{2}\ni \{x_n\}^{\infty}_{n=0} \rightarrow \sum^\infty_{n=0}\frac{x_n}{\sqrt{2^n}}\in \mathbb{K}$$ As for continuity, I've found a constant $C=\sqrt{2}$ (derived it from Holder inequality. But I don't know how to find norm of this functional. I can't find a vector from $\ell^{2}$ for which the inequality $$| \sum^\infty_{n=0}\frac{x_n}{\sqrt{2^n}} | \le \sqrt{2} \cdot(\sum^\infty_{n=0}|x_n|^2)^{1/2} $$ is an equality.

Thanks for any help.

Answer 1

If $(X,\langle,\rangle)$ is an inner product space, then for all $x,y\in X$, $$|\langle x,y\rangle|\leqslant \sqrt{\langle x,x\rangle}\cdot \sqrt{\langle y,y\rangle}$$ and there is equality if and only if $x=\alpha y$ for some scalar $\alpha$.