Sometimes on a single vector space, different norms can be defined. In this post, first we give the definition of an inner product vector space. Then, we proceed to discuss different norms on the finite-dimensional real vector spaces, i.e. the vector spaces of the form \(\mathbb{R}^n\).

#### The dot product

In the post on the dot product of vectors, we have discussed the dot product of the vectors spaces \(\mathbb{R}^2\) and \(\mathbb{R}^3\). In the post on the finite-dimensional vectors, we have discussed the dot product of finite-dimensional real vectors.

The usual dot product in \(\mathbb{R}^n\) can be interpreted as a matrix multiplication of two matrices. More precisely, if $$ u = (u_1, \dots, u_n)$$ and $$ v = (v_1, \dots, v_n),$$ then the dot product of the vectors \(u\) and \(v\) is the multiplication of the matrix $$\begin{pmatrix} u_1 & \cdots & u_n \end{pmatrix}$$ with the transpose of matrix $$\begin{pmatrix} v_1 & \cdots & v_n \end{pmatrix}.$$ Note that $$u \cdot v^T$$ is computed as follows: $$\begin{pmatrix} u_1 & \cdots & u_n \end{pmatrix} \cdot \begin{pmatrix} v_1 \\ \vdots\\ v_n \end{pmatrix} = $$ $$\sum_{i=1}^{n} u_i v_i.$$

### Euclidean norm

As we have explained in the post on the finite-dimensional vectors, the Euclidean norm of a vector, denoted by \(\Vert \cdot \Vert\), is the square root of the dot product of a vector to itself, i.e., $$\Vert u \Vert = \sqrt{u \cdot u}.$$

### Inner product real vector spaces

A real vector space \(V\) is called to be an inner product vector space if there is a function of the form $$ V \times V \rightarrow \mathbb{R}$$ defined by $$(u,v) \mapsto \langle u , v \rangle $$ such that the following rules hold for all vectors \(u\), \(v\), and \(w\) and real numbers \(r\) and \(s\):

- \(\langle u , v \rangle = \langle v , u \rangle\), (symmetry).
- \( \langle ru+sv , w \rangle = \) \(r \langle u , w \rangle + s \langle v , w \rangle\), (linearity).
- If \(u\) is nonzero, then \(\langle u , u \rangle > 0\), (positive definiteness).

Now, we proceed to define normed vector spaces.

### Real vector spaces with norms

A real vector space \(V\) is a normed vector space if there is a function $$\nu : V \rightarrow \mathbb{R}$$ with the following properties:

- \(\nu(u) \geq 0\), and if it happens that \(\nu(u) = 0\), then \(u = \vec{0}\), for all \(u\) in \(V\).
- \(\nu(ru) = \vert r \vert \nu(u)\), for all \(u\) in \(V\) and real number \(r\).
- \(\nu(u + v) \leq \nu(u) + \nu(v)\), for all \(u\) and \(v\) in \(V\).

A real vector space with an inner product has at least one norm or as mathematicians say any inner product induces a norm called its canonical norm. More precisely, if $$ V \times V \rightarrow \mathbb{R}$$ defined by $$(u,v) \mapsto \langle u , v \rangle $$ is the inner product of the real vector space \(V\), then its canonical norm is defined by $$\nu(u) = \sqrt{\langle u , u \rangle}.$$

### Different norms of the real vector spaces \(\mathbb{R}^n\)

In the real vector space \(\mathbb{R}^n\), the Euclidean norm, which is obtained from the dot product, measures the straight-line distance of a vector from the origin. However, for any real number \(p \geq 1\), one can define the so-called \(L_p\) norm as follows: If $$ u = (u_1, \dots, u_n),$$ then $$\Vert u \Vert_p = \left( \sum_{i=1}^{n} \vert u_i \vert^{p}\right)^{1/p}.$$

The \(L_1\) norm is called Taxicab norm and simply calculated by adding the absolute value of the components of a vector. In other words, $$\Vert u \Vert_1 = \sum_{i=1}^{n} \vert u_i \vert.$$ The name is derived from this fact that in a city with a “grid plan”, a taxi needs to drive in a “rectangular street grid” to get from the origin to the point \(u\).

If \(p\) approaches to infinity, then the \(L_p\) norm approaches to the infinity norm obtained by the following formula: $$\Vert u \Vert_{\infty} = \max_{i} \vert u_i \vert.$$ For this reason the infinity norm is called the maximum norm, in some references.

### Comparison of \(L_p\) norms

It is an easy exercise to see that for all \(u\) in \(\mathbb{R}^n\), the following inequalities hold: $$\Vert u \Vert_2 \leq \Vert u \Vert_1 \leq \sqrt{n} \Vert u \Vert_2.$$

As a matter of fact, if \(p\geq 1\) and \(\alpha \geq 0\) are real numbers, for all \(u\) in \(\mathbb{R}^n\), the following inequalities hold: $$\Vert u \Vert_{\infty} \leq \Vert u \Vert_{p+\alpha} \leq \Vert u \Vert_p.$$

**Example**. Let \(u = (5,6, – 30)\). Since $$5^2 + 6^2 + (-30)^2 = 961,$$ the Euclidean norm of \(u\) is $$\Vert u \Vert_2 = 31.$$ Also, the taxicab norm and the infinity norm of \(u\) are as follows:

- \(\Vert u \Vert_1 = 5+6+30 = 41.\)
- \(\Vert u \Vert_{\infty} = \max\{5,6,30\} = 30.\)

Finally, the \(L_3\) norm of \(u\) is as follows: $$\Vert u \Vert_3 = $$ $$\sqrt[3]{5^3+6^3+30^3} =$$ $$\sqrt[3]{27341} \approx 30.125.$$

### The distance in real vector spaces with norms

If the norm function of a real vector space is \(\nu\), then its distance function is defined as follows: $$d(u,v) = \nu(u-v).$$ By comparing different norms, one can easily deduce that the Euclidean distance between two vectors \(u\) and \(v\) in \(\mathbb{R}^n\) is, for example, not bigger than their taxicab distance.

**Exercise**. Prove that the function \(p: \mathbb{R}^2 \rightarrow \mathbb{R}\) defined by $$p(x,y) = \vert x \vert + 2 \vert y \vert$$ is a norm for \(\mathbb{R}^2.\)