MHB Why Does Vector Norm Use "Double" Absolute Value?

[SweatingBear]

Why is it that the norm of a vector is written as a "double" absolute value sign instead of a single one? I.e. why is the norm written as $ || \vec{v} || $ and not $ | \vec{v} | $? I think $ | \vec{v} | $ is appropriate enough, why such emphasis on $ || \vec{v} || $? I think it's rather natural to interpret the "absolute value" of a vector as its length (magnitude), just like in complex analysis.

 

[Ackbach]

The notation comes from functional analysis, where you have "vectors" that might be functions. In that case, $\| \, f \|$ might have a very different meaning from $ \left| \,f \right|$. In fact, $ \left| \,f \right|$ might have no meaning at all. One definition is the 1-norm:
$$ \| \, f \|_{1}:=\int_{X} \left| \, f \right| \, d\mu.$$
It might be confusing if you used single bars on the LHS of this definition.

 

[Deveno]

Another reason is this, we have the equation for a scalar [math]\alpha[/math] and a vector [math]v[/math]:

[math]\|\alpha v\| = |\alpha|\cdot \|v\|[/math]

where it makes sense to distinguish between the two types of "norms" being used on the vector space, and the underlying field.

Nevertheless, in many abstract treatments of linear algebra, only single vertical bars are used, with the double-vertical bar used for linear algebra with physical interpretations (where vectors and scalars represent different KINDS of entities).

In terms of complex numbers, the complex modulus turns out to BE the absolute value on the real line...the trouble is, in [math]\Bbb R^n[/math] there's no "natural" line through the origin to pick as "the real line" (there are certain exceptions for the special cases n = 1, 2, 4, 8 and 16, but these are too complicated to go into here).

 

[SweatingBear]

Fair enough, thanks!