# What does $\|u\|$ mean?

What does $\left\Vert \mathbf{u}\right\Vert$ mean in this equation? How would this equation be performed? I'm extremely terrible in discrete mathematics and a simplistic answer would be ideal. (Don't answer it directly, I want to practice) • It looks like you have to calculate the norm of the given vector. – Hirshy Aug 26 '15 at 8:05

## 2 Answers

The norm of a vector $(1,3,4,11,13)$ is $\sqrt{1^2+3^2+4^2+11^2+13^2}$. It is an extension of Pythagoras' Theorem.

• For this you're assuming that we have the euclidean norm. It is at least worth mentioning that there are other norms which one has to deal with in a different way. – Hirshy Aug 26 '15 at 8:08
• Surprisingly this was the solution. Thank you Michael. – Oliver K Aug 26 '15 at 8:34
• Is this what people mean when they call it the "Euclidian (L2) norm"? – Mark White Nov 30 '18 at 18:37
• Yes, that's right. – Empy2 Nov 30 '18 at 18:58
• @Hirshy what are the other kind of norms different from this one? – bikalpa Feb 26 '19 at 13:44

To elaborate on my comment on Michael's answer:

The symbol $\left\Vert\mathbf{u}\right\Vert$ for a vetor $\mathbf{u}$ usually stands for the norm of that vector. A norm is "a function that assigns a strictly positive length or size to each vector in a vector space" (quoted from wikipedia).

Having a normed vector space enables you to talk about e.g. the length of a vector. A common example would be the vector space $\mathbb R^n$ with the euclidean norm which is the norm induced by the dot product $\langle v,w\rangle = \sum\limits_{i=1}^n v_iw_i$ where $v=\begin{pmatrix} v_1 \\ \vdots \\ v_n \end{pmatrix},w=\begin{pmatrix} w_1 \\ \dots \\ w_n\end{pmatrix}$, but there are other norms and inner products one can use.

Let $\mathcal V$ be a vector space with $\dim(\mathcal V)=n$ over $\mathbb R$ and $\langle\cdot,\cdot\rangle$ an inner product on $\mathcal V$. Then $(\mathcal V,\left\Vert\cdot\right\Vert)$ is a normed vector space with $\left\Vert v\right\Vert:= \sqrt{\langle v,v\rangle}$. The norm of a vector can be interpreted as the length of the vector but it is dependent on which inner product one uses.

• Thanks Hirshy. A great informative post. +1 – Oliver K Aug 26 '15 at 9:59