Problem I'm trying to show that the canonical dual pairing $(\cdot, \cdot):V^{\vee} \times V \rightarrow \mathbb{F}$ between a normed vector space $(V, \mathbb{F})$ and dual $V^{\vee}$ defined by $(f, v) := f(v)$ for all $f \in V^{\vee}$ and $v \in V$ is continuous.
Thoughts The dual paring is a bilinear function; if I can show that $(\cdot, \cdot)$ carries bounded sets to bounded sets, this will suffice since any $n-$linear function that satisfies this criterion is necessarily continuous. It is tempting to write $ |(f, v)| = |f(v)| \leq |f|_*|v|_V $ where $|\cdot|_*$ denotes the operator norm on $V^{\vee}$ and $|\cdot|_V$ denotes the norm on $V$. If this equation were true, holding $v$ fixed would provide a bound for $f$ and holding $f$ fixed would provide a bound for $v$. But, the operator norm is only well-defined for bounded linear functions and I'm not certarin that a linear functional is necessarily bounded.
Question So, am I on the right track? If not, how should I approach this problem?