Normed Spaces Over the Field of Real Numbers

# Normed Spaces Over the Field of Real Numbers

 Definition: Let $V$ be a vector space over the field $\mathbb{R}$. A Norm on $V$ is a function $\| \cdot \| : V \to [0, \infty)$ that satisfies the following properties: 1) $\| x \| \geq 0$ for all $x \in V$ and $\| x \| = 0$ if and only if $x = 0$. 2) $\| \alpha x \| = \mid \alpha \mid \| x \|$ for all $x \in V$ and for all $\alpha \in \mathbb{R}$. 3) $\| x + y \| \leq \| x | + \| y \|$ for all $x, y \in V$.

Of course, a norm can be defined on vector spaces over other fields, such as $\mathbb{C}$.

 Definition: A Normed Space $(V, \| \cdot \|)$ is a vector space $V$ with a norm $\| \cdot \|$ defiend on $V$.

For example, consider the vector space $\mathbb{R}^n$. Then the common Euclidean norm on $\mathbb{R}^n$ is defined for all $\mathbf{x} = (x_1, x_2, ..., x_n) \in \mathbb{R}^n$ by:

(1)
\begin{align} \quad \| \mathbf{x} \| = \sqrt{x_1^2 + x_2^2 + ... + x_n^n} \end{align}

Clearly property (1) . For property (2), let $\alpha \in \mathbb{R}$ and let $\mathbf{x} \in \mathbb{R}^n$. Then:

(2)
\begin{align} \quad \| \alpha \mathbf{x} \| &= \sqrt{(\alpha x_1)^2 + (\alpha x_2)^2 + ... + (\alpha x_n)^n} \\ \quad &= \sqrt{\alpha^2(x_1^2 + x_2^2 + ... + x_n^2)} \\ \quad &= \sqrt{\alpha^2}\sqrt{x_1^2 + x_2^2 + ... + x_n^2} \\ \quad &= \mid \alpha \mid \| \mathbf{x} \| \end{align}

The third property is left for the reader to verify.