Show that any bounded linear functional on a normed linear space is continuous Show that any bounded linear functional on a normed linear space is continuous. Can we say that it is uniformly continuous ? Also, is it true, if we reverse the statement any continuous linear functional on a normed linear space is bounded ??

Let $L:X\to\mathbb R$ be a bounded linear functional on a normed space $X$. Then there exists some $M>0$ such that $|L(x)|\leq M||x||$ for each $x\in X$. To show that $L$ is continuous note that $|L(x)-L(y)|=|L(x-y)|\leq M||x-y||$ for each $x,y\in X$. In particular, $L$ is even uniformly continuous.

For the converse assume that $L$ is continuous but not bounded. Then there exists for each $n\in\mathbb N$ a $x_n\in X$ such that $|L(x_n)|>n||x_n||$. In particular, $x_n\
eq0$. This implies $$ 1\leq\frac{|L(x_n)|}{n|||x_n||}=\left|L\left(\frac{x_n}{n||x_n||}\right)\right|\to|L(0)|=0, $$ a contradiction. This limit comes from the fact that $L$ is continuous (at 0), and $\frac{x_n}{n||x_n||}\to0$.