Is this linear functional bounded? Find the norm. $$\ell^2\ni (x_n)\rightarrow2x_{1}+28x_2+35 x_{3}$$
I think it can be bounded:
$$|2x_{1}+28x_2+35 x_{3}| \le |2x_{1}|+|28x_2|+|35 x_{3}| \le 65 (\sum_{n=0}^{\infty}|x_n|^2)^{1/2}$$
But I can't find norm.

What is the norm of the linear functional $L(x)$? It is the smallest constant $M$ such that
$$|L(x)| \leq M ||x||_2$$
holds for every $x \in \ell^2$.
Okay, so now note that your linear functional is of the form $\langle a, x \rangle$, so apply the Cauchy inequality to get
$$|L(x)| = |\langle a, x \rangle| \leq ||a||_2 ||x||_2.$$
This means $M \leq ||a||_2$. Now use the fact that $a_1 x_1 + a_2 x_2 + a_3 x_3 = ||a||_2 ||x||_2 \cos \theta$, provided $x_n = 0$ for $n>3$, to show that if $M < ||a||_2$, you could pick an $\theta$ and hence an $x$, such that $|\langle a, x \rangle| > M ||a||_2 ||x||_2$. 
