Prove that $S_{n} : X \to X$ is continuous mapping.

Question

Consider $X$ - Banach space with Schauder basis : $\forall x \in X$ $\exists!$ $x = \sum_{i=1}^{\infty} x_{i}e_{i}$.

Let's consider $S_{n} = \sum_{i = 1}^{n} x_{i}e_{i}$. We want to show that it is a continous mapping (i.e. $\exists C > 0: $ $\|S_n\| \le C \|x\|$, for all $x \in X$).

How should I start with? Should I use Cauchy-sequences?

Answer 1

$x = \sum_{i=1}^\infty x_i e_i$ means that the partial sums converge to $x$ in norm. So, if you add and subtract $x$ inside $\|S_n\|$, then apply the triangle inequality, you should be able to finish the proof.