Norm

Definition: Norm

Let \(X\) be a space. A norm is a function \(f: X \rightarrow \mathbb{R}\) satisfying the three following properties for \(x_1, x_2 \in X\) and for \(\alpha \in \mathbb{R}\):

1. Positive-definiteness: \(f(x_1) \geq 0\) and \(f(x_1) = 0 \Leftrightarrow x_1 = 0\)

2. Absolute homogeneity: $$f(\alpha x_1) =

   \alpha

   f(x_1)$$

3. Triangle inequality: \(f(x_1 + x_2) \leq f(x_2) + f(x_2)\)

Normed Vector Space

A normed vector space is a vector space equipped with a norm. Frequently, if we are dealing with an inner product space (i.e. a vector space equipped with an inner product), we can define a norm as the square root of the inner product:

\[\lvert \lvert x \lvert \lvert_2 := \sqrt{ \langle x, x \rangle }\]