Let $T$ be a linear operator in $\mathcal{L}(V)$. An operator norm is denoted as $||T||$, where it is the smallest $M$, such that $||T(v)||$ $\le$ $M||v||$ for any $v \in V$.
A norm on the vector space $V$ is simply the square root of an inner-product.
Do these two types of norms have some relationship with each-other, and if so, what is it?
You are wrong in your assumption that “A norm on the vector space $V$ is simply the square root of an inner-product.” Most norms are not induced by inner-products. For instance, of all norms $\lVert\cdot\rVert_p$ in $\mathbb{R}^n$ defined by$$\bigl\lVert(x_1,\ldots,x_n)\bigr\rVert_p=\bigl(\lvert x_1\rvert^p+\cdots+\lvert x_n\rvert^p\bigr)^{\frac1p}$$($p\in[1,\infty)$), only the norm $\lVert\cdot\rVert_2$ is induced from an inner-product.