Proving a linear functional on a Banach space is continuous.

Question

Let $X$ be a Banach space and $Y$ be its proper closed subspace, i.e $Y \lneq X$. For some $x_0 \in X \setminus Y$, I constructed the following functional

$\gamma: Y \oplus \mathbb{R}x_0\to \mathbb{R}$ by defining $\gamma (y+tx_0) = t$.

It is easy to check $\gamma$ is linear, but I don't know how to show $\gamma$ is continuous. I'd appreciate it if you'd help me with this. Thank you in advance.

Answer 1

Projection onto a coordinate is always continuous. Remember, you want to show that given an open set in the codomain, you can find an open set in the domain which maps inside. Since your codomain is $\mathbb{R}$ you may assume you're looking at something like $(t_0-\epsilon,t_0+\epsilon)$. It is not hard to show that the open set you want in the domain is $Y\times (t_0-\epsilon,t_0+\epsilon)x_0$.

Answer 2

Or just use sequences: if $z_n:=y_n+t_nx_0\to 0$ then $t_n\to 0$ (because $x_0\notin Y$ and $Y$ is closed.) Then, $\gamma(z_n)=t_n\to 0.$