For finite-dimensional vector space $X$, does $\phi: X \rightarrow \mathbb{R}$ continuous and bounded imply $\phi$ linear?

Question

In a finite-dimensional vector space $X$, if $\phi: X \rightarrow \mathbb{R}$ is linear, then it can be shown to be continuous and bounded.

Bounded in this context means that there exists an $M < \infty$ such that for all $x\in X$, $|\phi(x)| \le M \|x\|$.

Does the converse hold? I.e. if we have a functional $\phi: X \rightarrow \mathbb{R}$ that is bounded and continuous, need it be linear?

Answer 1

No. For instance, consider $\phi(x)=\|x\|$.