Let $(X,||\cdot||)$ be a normed vector space and let $Y$ be a linear subspace of $X$. Let $f:Y\to \mathbb{R}$ be a continuous linear functional on $Y$. Let $F:X\to \mathbb{R}$ be a linear functional on $X$ such that $F=f$ on $Y$, i.e., $F(y)=f(y), \forall y\in Y$. Is $F$ continuous on $X$? Or is this not enough structure?
Note that I am NOT asking if an extension, $F$, that is continuous exists. Rather, I am trying to show a specific $F$ that has the above properties is continuous.