We know the Cauchy–Schwarz inequality for inner product space such that
\begin{align} |\langle u,v \rangle|^2 \le \langle u,u \rangle \langle v,v \rangle \end{align} Then if we induce the norm by $\|u\| := \sqrt{\langle u,u \rangle}$ we obtain \begin{align} |\langle u,v \rangle| \le \| u \| \| v \| \end{align}
On the other hand, for a continuous bilinear form $b(u,v)$ that is bounded we have \begin{align} b(u,v) \le C \| u \| \| v \| \end{align} which is very similar to the Cauchy–Schwarz inequality except a constant $C$.
I already know the inner product can been considered as a symmetric nonnegative bilinear form. While I do not yet find but I believe there are some inherent connections between the Cauchy–Schwarz inequality and the continuity/boundedness inequality. Can someone give me an answer?