Proving that a function $G: X\rightarrow \mathbb{R}^{n+1}$ is continuous.

Question

$(X, ||\cdot||)$ is a normed vector space, and $l_0, l_1, ..., l_n$ are linear operators in $X^*$. And $G$ is defined like this. $G:X\rightarrow R^{n+1}: x\rightarrow (l_0(x), l_1(x), ..., l_n(x))$. How do I prove that G is continuous? I've already proven that $G$ is linear but for continuity I'm a little lost. I'm trying by proving that a $L\geq 0$ exist such that $||G(x)|| \leq L\cdot||x||$, and applying that to each $l_i(x)$ because for each one, a different $L$ exits, given that they are continuous. So I thought that $||G(x)||$ could be bounded by the sum of all the different $L_i$ times $||x||$. Is that correct? All this cosidering the euclidian norm for $\mathbb{R}^{n+1}$

Answer 1

I assume $X^*$ is the continuous dual space. If that's the case, then it's a basic property of the product topology that a map $X \rightarrow \mathbb{R}^{n+1}$ is continuous iff its projection onto each factor is continuous. In other words, each of the maps $l_i : X \rightarrow \mathbb{R}$ are continuous, so $G$ is continuous.