Three Ways to Measure a Vector
A vector can have several reasonable measures of size. One measure adds the magnitudes of all its coordinates; another combines their squares; a third records only the largest coordinate magnitude. These choices answer different questions about how a vector’s size is distributed. For example, a vector with one large coordinate and a vector with many moderate coordinates can have the same total magnitude but different largest-coordinate sizes.
Throughout, let \(n\ge 1\) and let \(\mathbf{x}=(x_1,\dots,x_n)^{\top}\in\mathbb{R}^n\). The absolute value \(|x_i|\) measures the magnitude of the \(i\)th coordinate without regard to its sign. The Euclidean norm \(\|\mathbf{x}\|_2\) uses the usual square root of the sum of squared coordinates. We will compare it with two other standard measures.
For real coordinates, \(|x_i|^2=x_i^2\). The maximum in the infinity norm is taken over the finitely many coordinate magnitudes, so it is attained by at least one coordinate. The word “infinity” is part of the norm’s name; the formula does not require an infinite-dimensional vector or an infinite sum.
Each formula focuses on a different feature. The 1-norm accumulates every coordinate magnitude at the same rate. The 2-norm also uses every coordinate, but squaring makes a large coordinate contribute more strongly than a small one. The infinity norm ignores all but the largest magnitude. These are distinct ways to summarize the same list of coordinates, not interchangeable formulas.
Calculating the Norms
Worked Example: A Vector With Mixed Signs
Calculate all three norms of $$ \mathbf{x}=\begin{bmatrix}3\\-4\\1\end{bmatrix}. $$
For the 1-norm, take absolute values and add them:
For the 2-norm, square the coordinates, add, and take the nonnegative square root:
For the infinity norm, compare the absolute coordinate values:
Check: the defining formulas give \(8\), \(\sqrt{26}\), and \(4\), respectively. In particular, the negative sign on the second coordinate does not make its magnitude negative: its contribution is \(4\) to the 1-norm, \(16\) inside the square root for the 2-norm, and \(4\) among the candidates for the maximum.
Worked Example: Concentrated and Distributed Coordinates
Compare the norms of the vectors $$ \mathbf{a}=\begin{bmatrix}6\\0\\0\end{bmatrix}, \qquad \mathbf{b}=\begin{bmatrix}2\\2\\2\end{bmatrix}. $$
For \(\mathbf{a}\), the coordinate magnitudes are \(6,0,0\). Thus $$ \|\mathbf{a}\|_1=6+0+0=6,\qquad \|\mathbf{a}\|_2=\sqrt{6^2+0^2+0^2}=\sqrt{36}=6,\qquad \|\mathbf{a}\|_{\infty}=\max\{6,0,0\}=6. $$
For \(\mathbf{b}\), the coordinate magnitudes are \(2,2,2\). Thus $$ \|\mathbf{b}\|_1=2+2+2=6,\qquad \|\mathbf{b}\|_2=\sqrt{2^2+2^2+2^2}=\sqrt{12}=2\sqrt{3},\qquad \|\mathbf{b}\|_{\infty}=\max\{2,2,2\}=2. $$
Check: both 1-norms are \(6\), since each vector has total coordinate magnitude \(6\). But \(\mathbf{a}\) puts all of that magnitude in one coordinate, so its infinity norm is \(6\); \(\mathbf{b}\) distributes it evenly, so its infinity norm is \(2\). The 2-norms also differ: \(6\) for \(\mathbf{a}\), and \(2\sqrt{3}\) for \(\mathbf{b}\).
Worked Example: A Coordinate That Controls the Maximum
Calculate the norms of $$ \mathbf{z}=\begin{bmatrix}-2\\5\\-1\\0\end{bmatrix}. $$
The 1-norm adds all four absolute values: $$ \|\mathbf{z}\|_1=|-2|+|5|+|-1|+|0|=2+5+1+0=8. $$ The 2-norm is $$ \|\mathbf{z}\|_2 =\sqrt{(-2)^2+5^2+(-1)^2+0^2} =\sqrt{4+25+1+0} =\sqrt{30}. $$ The largest absolute coordinate is \(5\), so $$ \|\mathbf{z}\|_{\infty}=\max\{2,5,1,0\}=5. $$
Check: the largest coordinate magnitude, \(5\), contributes \(5\) to the 1-norm and \(25\) under the square root in the 2-norm. The remaining nonzero coordinates contribute \(3\) more to the 1-norm and \(5\) more under that square root, producing \(8\) and \(\sqrt{30}\).
Why These Formulas Are Norms
A norm on \(\mathbb{R}^n\) is a function \(\|\cdot\|\) from vectors to nonnegative real numbers that satisfies three properties: \(\|\mathbf{x}\|=0\) exactly when \(\mathbf{x}=\mathbf{0}\); \(\|c\mathbf{x}\|=|c|\|\mathbf{x}\|\) for every real scalar \(c\); and \(\|\mathbf{x}+\mathbf{y}\|\le \|\mathbf{x}\|+\|\mathbf{y}\|\) for every pair of vectors. The last property is the triangle inequality. The formulas above meet all three requirements.
Proof. First consider nonnegativity and the zero condition. Each absolute value is nonnegative, so each of the three formulas is nonnegative. If \(\|\mathbf{x}\|_1=0\), then the sum of the nonnegative numbers \(|x_i|\) is zero, which forces every \(|x_i|=0\), hence every \(x_i=0\). If \(\|\mathbf{x}\|_2=0\), its square is \(\sum_{i=1}^n x_i^2=0\); since each square is nonnegative, every \(x_i=0\). If \(\|\mathbf{x}\|_{\infty}=0\), then every \(|x_i|\le 0\), so every \(x_i=0\). Conversely, substituting \(\mathbf{x}=\mathbf{0}\) into any formula gives zero.
For homogeneity, let \(c\in\mathbb{R}\). Since \(|cx_i|=|c||x_i|\), $$ \|c\mathbf{x}\|_1=\sum_{i=1}^n|cx_i|=|c|\|\mathbf{x}\|_1. $$ Also, $$ \|c\mathbf{x}\|_2 =\left(\sum_{i=1}^n c^2x_i^2\right)^{1/2} =|c|\left(\sum_{i=1}^n x_i^2\right)^{1/2} =|c|\|\mathbf{x}\|_2. $$ Taking the maximum of the equalities \(|cx_i|=|c||x_i|\) gives \(\|c\mathbf{x}\|_{\infty}=|c|\|\mathbf{x}\|_{\infty}\). These equations also cover \(c=0\).
For the triangle inequalities, let \(\mathbf{x},\mathbf{y}\in\mathbb{R}^n\). The scalar inequality \(|x_i+y_i|\le |x_i|+|y_i|\), summed over \(i\), gives $$ \|\mathbf{x}+\mathbf{y}\|_1\le \|\mathbf{x}\|_1+\|\mathbf{y}\|_1. $$ For the 2-norm, use the Cauchy–Schwarz inequality for the Euclidean dot product, \(|\mathbf{x}^{\top}\mathbf{y}|\le\|\mathbf{x}\|_2\|\mathbf{y}\|_2\), from the earlier material on inner products. Expanding the square gives $$ \|\mathbf{x}+\mathbf{y}\|_2^2 =\|\mathbf{x}\|_2^2+2\mathbf{x}^{\top}\mathbf{y}+\|\mathbf{y}\|_2^2 \le \bigl(\|\mathbf{x}\|_2+\|\mathbf{y}\|_2\bigr)^2. $$ Both sides before squaring are nonnegative, so taking square roots proves the 2-norm triangle inequality. Finally, for each coordinate \(i\), $$ |x_i+y_i|\le |x_i|+|y_i| \le \|\mathbf{x}\|_{\infty}+\|\mathbf{y}\|_{\infty}. $$ Taking the maximum over \(i\) proves the infinity-norm triangle inequality. All three functions satisfy the norm properties. \(\square\)
Comparing the Three Sizes
The norms have different formulas, but they cannot differ without limit when the dimension \(n\) is fixed. The following inequalities give precise comparisons. They also help interpret the calculations: no one of the three measures can be arbitrarily large or small compared with another for vectors in the same \(\mathbb{R}^n\), although the bounds involve \(n\).
Proof. Let \(M=\|\mathbf{x}\|_{\infty}\). Each coordinate satisfies \(|x_i|\le M\), so \(x_i^2\le M^2\). In particular, any one coordinate square is no greater than the sum of all the coordinate squares. Taking the largest coordinate magnitude therefore gives \(\|\mathbf{x}\|_{\infty}\le\|\mathbf{x}\|_2\). Summing \(x_i^2\le M^2\) over the \(n\) coordinates gives $$ \|\mathbf{x}\|_2^2=\sum_{i=1}^n x_i^2\le nM^2=n\|\mathbf{x}\|_{\infty}^2, $$ and taking nonnegative square roots proves \(\|\mathbf{x}\|_2\le\sqrt{n}\|\mathbf{x}\|_{\infty}\).
Next, expand the square of the 1-norm: $$ \|\mathbf{x}\|_1^2 =\left(\sum_{i=1}^n|x_i|\right)^2 =\sum_{i=1}^n x_i^2+2\sum_{1\le i<j\le n}|x_i||x_j| \ge \sum_{i=1}^n x_i^2 =\|\mathbf{x}\|_2^2. $$ Both norms are nonnegative, so \(\|\mathbf{x}\|_2\le\|\mathbf{x}\|_1\). For the other bound, apply Cauchy–Schwarz to the vectors \((|x_1|,\dots,|x_n|)^{\top}\) and \((1,\dots,1)^{\top}\): $$ \|\mathbf{x}\|_1 =\sum_{i=1}^n |x_i| \le \left(\sum_{i=1}^n x_i^2\right)^{1/2} \left(\sum_{i=1}^n 1^2\right)^{1/2} =\sqrt{n}\,\|\mathbf{x}\|_2. $$ This proves all four inequalities. \(\square\)
The factors involving \(n\) matter. The theorem does not say that all three numbers are equal; it bounds their relative sizes in a fixed dimension. In the first worked example, \(n=3\), and the values \(4,\sqrt{26},8\) satisfy the comparisons. In the second example, the equality \(\|\mathbf{a}\|_1=\|\mathbf{b}\|_1\) coexists with different infinity norms, illustrating that one measure alone does not reveal how the magnitude is distributed among coordinates.
Choosing and Interpreting a Norm
When describing a vector, the appropriate norm depends on which feature of its coordinates matters. The 1-norm reports cumulative magnitude, so many nonzero coordinates can add substantially even when none is large. The 2-norm is the familiar Euclidean length and gives a squared contribution to each coordinate before taking the square root. The infinity norm reports the largest individual coordinate magnitude; smaller coordinates do not change it unless they exceed the current maximum.
These distinctions are useful when comparing vectors with the same number of coordinates. For instance, the pair \(\mathbf{a},\mathbf{b}\) above has equal 1-norms but different infinity norms. Conversely, two vectors can share an infinity norm while having different totals: \((5,0,0)^{\top}\) and \((5,1,1)^{\top}\) both have infinity norm \(5\), whereas their 1-norms are \(5\) and \(7\). The maximum alone does not record what happens outside its largest coordinate.
There is no universally best choice among these measures. A calculation should state which norm is being used, especially when a phrase such as “the size of the vector” could mean more than one thing. Once chosen, use that same definition consistently: absolute values belong in the 1-norm and infinity norm, and the square root belongs in the 2-norm.
Common Mistakes
Ignoring absolute values in the infinity norm. For \(\mathbf{x}=(3,-4,1)^{\top}\), a tempting but incorrect calculation is \(\max\{3,-4,1\}=3\). This takes the largest signed entry, not the largest coordinate magnitude. The definition requires $$ \|\mathbf{x}\|_{\infty}=\max\{|3|,|-4|,|1|\}=\max\{3,4,1\}=4. $$ The sign of a coordinate does not make its size negative.
Stopping before taking the square root. For the same vector, \(\sum_i x_i^2=9+16+1=26\), but \(26\) is the sum of squares, not the 2-norm. The required final step is \(\|\mathbf{x}\|_2=\sqrt{26}\). In contrast, the 1-norm uses a sum of absolute values, not a sum of squares: it is \(3+4+1=8\).
| Norm | What it uses | What it highlights |
|---|---|---|
| 1-norm | Sum of all coordinate magnitudes | Total accumulated magnitude |
| 2-norm | Square root of the sum of squared coordinates | Euclidean length, with large coordinates contributing strongly |
| Infinity norm | Largest coordinate magnitude | The single largest component |
Check Your Understanding
Use the definitions directly, and include absolute values wherever the formula requires them.
- For \(\mathbf{u}=(1,-2,2)^{\top}\), calculate \(\|\mathbf{u}\|_1\), \(\|\mathbf{u}\|_2\), and \(\|\mathbf{u}\|_{\infty}\).
- Find the three norms of \(\mathbf{v}=(-3,0,4,0)^{\top}\). Which coordinate determines its infinity norm?
- Give two vectors in \(\mathbb{R}^3\) with the same 1-norm but different infinity norms, and verify both claims.
- For \(\mathbf{w}\in\mathbb{R}^5\) with \(\|\mathbf{w}\|_{\infty}=2\), state the upper bound on \(\|\mathbf{w}\|_2\) supplied by the comparison theorem.
- Explain why \(\max_i x_i\) is not the definition of the infinity norm for a vector that may have negative coordinates.
