Let $t_1,\ldots,t_n\in[a,b]$, $c_1,\ldots,c_n\in\mathbb C$ and define a linear functional $f(x)=\sum_{j=1}^nc_jx(t_j)$ on $X$, where $X=C([-1,1], \mathbb C)$, the complex-valued continuous functions on the finite symmetric interval $[-1,1]$ of $\mathbb R$. Prove that $f$ belongs to $B(X,\mathbb C)$ and evaluate $\Vert f \Vert$.
This is a homework question from my functional analysis class.
In my opinion, to prove that $f$ belongs to $B(X,\mathbb C)$, firstly we have to show that $f$ is linear and secondly $f$ is bounded which is to show that $f$ is continuous.
I am just wondering if my fundamental method is correct?