Norms

Let $V$ be a real vector space. A function $\|\cdot\|: V \to \mathbb{R}$ is a norm if for all $x,y \in V$ and $\alpha \in \mathbb{R}$:

1. Non-negativity: $\|x\| \ge 0$ and $\|x\| = 0 \iff x = 0$
2. Homogeneity: $\|\alpha x\| = |\alpha|\,\|x\|$
3. Triangle inequality: $\|x+y\| \le \|x\| + \|y\|$

These axioms say length is never negative, scaling a vector scales its length by the same amount, and taking a two-step path cannot be shorter than the direct path.

Common norms on $\mathbb{R}^n$:

• $\ell_2$: $\|x\|_2 = \sqrt{\sum_i x_i^2}$
• $\ell_1$: $\|x\|_1 = \sum_i |x_i|$
• $\ell_\infty$: $\|x\|_\infty = \max_i |x_i|$

More generally, for $p \ge 1$,

$$\|x\|_p = \left(\sum_{i=1}^n |x_i|^p\right)^{1/p}.$$

If your norm comes from a dot product, then

$$\|x\| = \sqrt{x\cdot x}.$$

For the standard dot product on $\mathbb{R}^n$, this gives the Euclidean norm:

$$\sqrt{x\cdot x} = \sqrt{\sum_{i=1}^n x_i x_i} = \sqrt{\sum_{i=1}^n x_i^2} = \|x\|_2.$$

Not every norm comes from a dot product. The $\ell_1$ and $\ell_\infty$ norms are valid lengths, but they do not come from an inner product in the same way. This matters because dot-product geometry gives angles and projections, while general norms mainly give size and distance.

Once a norm is chosen, it induces a distance:

$$d(x,y) = \|x-y\|.$$

So changing the norm changes both what counts as a small vector and what counts as two points being close.