Norm and normed vector space

Let $X$ be a vector space over $\mathbb{R}$ (or $\mathbb{C}$). A norm on $X$ is a function $\|\cdot\|:X\to[0,\infty)$ such that for all $x,y\in X$ and scalars $\alpha$:

1. Positive definiteness: $\|x\|=0$ iff $x=0$.
2. Absolute homogeneity: $\|\alpha x\|=|\alpha|\,\|x\|$.
3. Triangle inequality: $\|x+y\|\le \|x\|+\|y\|$.

A pair $(X,\|\cdot\|)$ is called a normed vector space.

Context. Norms provide a quantitative notion of size. Via the induced metric , normed spaces are a fundamental class of metric spaces used to define convergence, continuity, and completeness.

Examples:

• On $\mathbb{R}^n$, $\|x\|_2=\big(\sum_{i=1}^n x_i^2\big)^{1/2}$ is a norm (Euclidean norm).
• On $\mathbb{R}^n$, $\|x\|_1=\sum_{i=1}^n |x_i|$ and $\|x\|_\infty=\max_i |x_i|$ are norms.
• On the space $C([0,1])$ of continuous functions, $\|f\|_\infty=\max_{t\in[0,1]}|f(t)|$ defines a norm.