$|x|$ versus $\lVert x\rVert$?

Question

I've taken linear algebra and currently reading through Principles of Mathematical Analysis by Walter Rudin. I noticed that he has the $\lVert x\rVert$ notation for the supremum of the L2 norms ( I think?) and also uses the $|x|$ notation as the absolute value. I'm having a hard time telling the difference while I got back and forth between his chapter on functions of several variables and my old linear algebra text by Farleigh and Beauregard. Does anyone have a guide to how I should interpret $|x|$ and $\lVert x\rVert$ depending on when the text was written? I'm looking for precise guides in the Rudin text but they might be alluding me so page numbers would help there if they exist.

Answer 1

Honestly, its a matter of convention. Usually $\lVert\:\cdot\:\rVert$ denotes a norm on a vector space, while $\lvert \cdot \rvert$ denotes the norm of a real number (i.e. its absolute value). It should also be understood as a norm.

Answer 2

The commenters and Antonios-Alexandros Robotis are correct: the two symbols are different ways of thinking about concepts which are basically the same. However, there are normal conventions which make some distinction between these notations.

• $\|\cdot\|$ almost always denotes a norm. The general idea is that if $V$ is a vector space over $\mathbb{R}$, then a norm on $V$ is a map $$ \|\cdot\| : V \to \mathbb{R}_{\ge 0} $$ which satisfies certain properties (e.g. $\|x\| = 0$ if and only if $x = 0$, $\|\lambda x\| = |\lambda| \|x\|$ for all $\lambda\in\mathbb{R}$, etc). Note that this can be generalized quite a bit (for example, $V$ needn't be a real vector space, and could, in principle, be a vector space over any valued field—but these are technical details which don't really matter here). The general notion is that a norm is a very specific notion of the size of a vector.
• $|\cdot|$ is a more general notation which can be used to describe size in a broader sense. For example:
  • If $x\in\mathbb{R}$, then $|x|$ denotes the absolute value of $x$, which can be thought of as the distance from $x$ to $0$ on the real number line. Note that $|\cdot|$ happens to be a norm on $\mathbb{R}$ (indeed, norms on vector spaces are a generalization of the absolute value on $\mathbb{R}$).
  • Similarly, if $z \in \mathbb{C}$, then $|z|$ denotes the modulus of $z$. The modulus function can also be interpreted as the distance from $z$ to $0$ in the complex plane. Again, this is a norm on $\mathbb{C}$.
  • More generally, if $k$ is any field and $(\Gamma, +, \le)$ is a totally ordered abelian group, then we might define a map $$ |\cdot| : k \to \Gamma_+ $$ which generalizes the notion of an absolute value. I'll not go into details here, but this gives rise to the notion of a valuation. Note that a valuation is not, in general, a norm (as the target space needn't be the real numbers).
  • The notation $|\cdot|$ also shows up in set theory. If $A$ is a set, then $|A|$ can be used to denote the cardinality of that set. For example, $$ |\{a,b,c\}| = 3, \qquad |\mathbb{N}| = \aleph_0, \qquad\text{and}\qquad |\mathbb{R}| = \mathfrak{c}. $$ Again, the notation $|\cdot|$ denotes some notion of size. Note that $\operatorname{card}(A)$ and $\#A$ might also be used in this context.
  • In contexts where there is no ambiguity, the notation $|\cdot|$ will often be used to denote a more general norm. For example, the text Stewart Calculus uses $|\mathbf{v}|$ to denote the norm of a vector in $\mathbb{R}^n$. This seems to occur more often in elementary calculus and linear algebra texts aimed at non-math majors, but is nevertheless another use of the notation.
• Happily enough, you might also see $\newcommand{\vvv}{|\!|\!|}\vvv\cdot\vvv$ in some contexts. Typically, this denotes a norm in a context where we might want to compare two different norms defined on the same vector space. For example, I seem to recall Munkres using this notation in his book on topology when discussing the notion of equivalent norms.

In short, $\|\cdot\|$ and $\vvv\cdot\vvv$ nearly always denote norms (I am not sure that I have ever seen these notations mean anything else). On the other hand $|\cdot|$ is typically used to denote some notion of size. In one sense it is a more specific notation (as the absolute value is a kind of ur-norm), but in another sense it is a more general notion, as valuation and cardinality (among other uses of the notation) give ideas of size which are distinct from a norm.

Answer 3

I'm not sure of this. But I think it is at least somewhat true. In the pre LaTek days, people had to type $\tt ||v||$ to say $\|v\|$ and I have see manuscripts where the author would type $\subset$ and manually write a line below to get $\subseteq$. If they had to do this A LOT, is was simpler to just write $\tt |v|$ and $\subset$. I personally prefer the minimalist approach. If the difference is not important to the subject at hand, then simpler notation is better. When the difference does matter, the authors can still use the more complicated symbols. They are allowed to as long as they are consisted throughout their article.