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 \(\displaystyle \alpha\) and a vector \(\displaystyle v\):

\(\displaystyle \|\alpha v\| = |\alpha|\cdot \|v\|\)

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 \(\displaystyle \Bbb R^n\) 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!

 

[blue_raver22]

The use of a "double" absolute value in the vector norm notation $|| \vec{v} ||$ is a convention that has been established in mathematics and has become a standard representation in scientific and engineering fields. This notation is used to emphasize the distinction between the absolute value of a scalar and the norm of a vector.

The absolute value of a scalar is a single value, whereas the norm of a vector is a measure of its length or magnitude. In other words, the norm of a vector is a scalar quantity, but it is not the same as the absolute value of a scalar. Therefore, using a single absolute value sign for the norm of a vector could lead to confusion and misinterpretation.

Moreover, the double absolute value notation for vector norms is also consistent with the notation used in other mathematical concepts, such as matrices and tensors. It allows for a more unified and systematic approach to representing mathematical operations involving vectors.

In addition, using the double absolute value notation for vector norms allows for a more intuitive understanding of the concept. Just like in complex analysis, where the absolute value of a complex number represents its distance from the origin, the norm of a vector represents its distance from the origin in a multi-dimensional space.

In conclusion, the use of a "double" absolute value in the vector norm notation is a convention that has been established for clarity, consistency, and intuitive understanding of the concept. It is a standard representation that is widely accepted and used in scientific and engineering fields.