Textbook
A norm on $\mathbb R^d$ is a function $\|\cdot\|:\mathbb R^d\to[0,+\infty)$ satisfying, for all $u,v\in\mathbb R^d$ and $c\in\mathbb R$, $\|v\|=0\Leftrightarrow v=0$, $\|cv\|=|c|\|v\|$, and $\|u+v\|\le\|u\|+\|v\|$. These require positive definiteness, consistent scaling, and the triangle inequality. The chosen norm induces distance $\|u-v\|$.
For real $p\ge1$, $\|v\|_p=(\sum_i|v_i|^p)^{1/p}$. In particular, $\|v\|_1=\sum_i|v_i|$, $\|v\|_2=\sqrt{\sum_iv_i^2}$, and $\|v\|_\infty=\max_i|v_i|$. An omitted subscript needs a convention; here it means the Euclidean norm $\|v\|_2$.
The notation $\|v\|_0$ often counts nonzero coordinates, but this is not a norm: doubling a nonzero vector does not double that count. A name cannot replace the axioms.
Absolute homogeneity handles negative scaling: taking $c=-1$ gives $\|-v\|=\|v\|$. The triangle inequality bounds the length of a sum from above without asserting equality; opposite vectors may cancel. Every axiom concerns one chosen measuring rule. Measuring one vector with norm one and another with norm two does not permit applying the axiom to those mixed numbers as though the subscript were the same throughout.
Definitions and further reading: Mathematics for Machine Learning.
In plain language
A norm fixes a ruler for measuring a vector: total coordinate movement, straight-line length, and the largest coordinate change give different rules. “Nearest” therefore leaves the agreement unfinished unless a norm is specified. Geometry has several tariffs, but switching them halfway through a comparison is not allowed. The road analogy explains norms 1 and 2, while the infinity norm measures the largest coordinate magnitude rather than a traveled route.
Example
For the original illustrative vector $v=(3,-4)^{\mathsf T}$, the three norms yield lengths 7, 5, and 4. The vector stays fixed; only the measuring rule changes. Each formula displays its intermediate quantities.
Vector Norms: Choosing a Notion of Length
Norm 1
The sum of absolute coordinates is 3 + 4 = 7.
Step 1 of 3: Norm 1
Read all steps
Norm 1
The sum of absolute coordinates is 3 + 4 = 7.
- State
- |3| = 3current|−4| = 4current7current
Norm 2
The square root of the sum of squares is 5.
- State
- 3² = 9current(−4)² = 16current√25 = 5current
Norm ∞
The largest absolute coordinate is 4.
- State
- 34includedmax: 4current
Across all three steps, the negative coordinate loses its sign through an absolute value or square, so length cannot become negative. In the Euclidean case, taking the square root of 25 is essential: the squared total 25 is not norm two itself. Doubling the original vector produces 14, 10, and 8 under the three rules. That check agrees with each rule’s homogeneity and helps reveal a missing absolute value or square root.
Related concepts: Absolute value, Coordinate vectors.