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?

1 Answer

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.

• Thanks for clearing this up for me. I'm in an introductory class as you might have guessed, and this is how it was introduced to us at this point. I'm reading Axler's Linear Algebra Done Right. – Jaigus Dec 10 '18 at 7:36