# Is there a special relationship between a norm on a vector space V, and the operator norm $\mathcal{L}(V, \mathbb{R)}$?

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.