Real analysis Math

Question

Prove the following:

Let $T$ be the linear operator from the normed space $X$ over $\mathbb{R}$ to the normed space $Y$ over $\mathbb{R}$. If $T$ is continuous on unit ball ($B(0,1)$) of $X$, then $T$ is continuous on $X$.

How can I prove this?

Answer 1

Let $x_0 \in X$ and let $(x_n)$ be a sequence in $X$ with $x_n \to x_0$. Then

$z_n:= x_n - x_0 \to 0$. Hence there ist $N \in \mathbb N$ such that $z_n \in B(0,1)$ for $n > N$. Since $T$ is continuous in $0$, we get $Tz_n \to T0=0$, hence

$Tx_n \to Tx_0$.