About the linearity of a function Let $X$ be a complex vector space. Let $Y$ be a dense subspace of $X$. Let $\phi:X\longrightarrow\mathbb{C}$ be a map such that $\phi$ is linear and bounded on $Y$. Can we say that $\phi$ is linear or continuous ?
Hahn Banach theorem only talks about the existence of a unique extension, it may not be the same map.

No. Define a function $f$ that takes $0$ on $Y$ and a constant non-zero vector on $X \setminus Y$. Then $f$ is neither continuous or linear.
