Definition of Vector norm

The inner products relays two types of information about vectors: their length or magnitude, and their relative orientation. While this may be difficult to visualize in $\mathbb{R}^{n}$, you won't be betrayed by visualizing $\mathbb{R}^{3}$. We solidify the notion of length in this manner. We define the Euclidean norm of a vector $\mathbf{x}$ in $\mathbb{R}^{n}$ as

$$\Vert \mathbf{x}\Vert _{2} = (\sum_{j=1}^{n} x_{j}^{2})^{1/2}.$$ (4)

By using this definition with that of the inner product we see that

$$(\mathbf{x},\mathbf{x}) = \Vert \mathbf{x}\Vert^{2}$$ (5)

The norm (and therefore the inner product) measures the length, size, magnitude, or strength of the vector depending on what interpretation you are giving the vector.

There is nothing special about the Euclidean norm. It is properties make it special. We recognize them as

As with the inner product, any function which takes a vector into a real number and satisfies these properties is called a norm. Item number (4) is refereed to as the triangle inequality.

We have already seen that inner products and norms are related. Another such relation is in the form of the Cauchy-Schwarz Inequality which says that for any two vectors $\mathbf{x}$ and $\mathbf{y}$

$$(\mathbf{x},\mathbf{y})^{2} \leq \Vert \mathbf{x}\Vert^{2} \Vert \mathbf{y}\Vert^{2}$$ (6)

Showing that a function you believe to measure size is indeed a norm can be a tricky proposition. In fact, showing the Euclidean norm is a actually a norm requires the use of the Cauchy-Schwarz Inequality to prove the Triangle Inequality.

If you have never seen norm proved to be so, it is worth a quick peek.