So the question would be:What is the difference between absolute value and norm?

[lampshade]

Hello,
I don't know if this is general math or not, but I know it is a simple question that I just don't get so I put it in this forum.


I'm kinda rusty at this, but I came across an equation in a book on wavelets that uses the notation

| f(t) |

inside an integral and I"m not sure what they mean. What does the | | notation mean? Do they mean absolute value? Are they using it to mean magnitude but just not using the || a || notation I'm more familiar with?

they use it in an integral from negative infinity to infinity of |f(t)|dt

thanks in advance,
Lampshade

 

[lampshade]

After studying it some more, I think they do actually mean the absolute value and nothing else. I'm still a bit shaky though so respond with insights if you have any, please.

 

[cristo]

Yes, it means absolute value.

 

[k3N70n]

In general you should be able to tell from the context which is meant. Many books with use the same notation for absolute value as well as magnitude.

 

[Integral]

In general you should be able to tell from the context which is meant. Many books with use the same notation for absolute value as well as magnitude.

Really, because they are the same. The Absolute value can be seen as the magnitude of the number.

 

[HallsofIvy]

In general you should be able to tell from the context which is meant. Many books with use the same notation for absolute value as well as magnitude.

Really, because they are the same. The Absolute value can be seen as the magnitude of the number.

That is if f is a numerical function. I think k3N70n was referring to the situation where f is a vector function. Many textbooks use | | for "magitude" (length) of a vector as well as absolute value of a number.

 

[ObsessiveMathsFreak]

||x|| generally means the two norm. |x| can also be the two norm, but sometimes it stands for something else.

 

[Xevarion]

You should never assume $$\|x\|$$ is the $$L^2$$ norm, unless the author has stated that explicitly. It just means 'the norm of x'. Which norm should be clear from the context; if not, go back a few pages and try to figure it out.

I don't think I have ever seen |x| used for any norm other than the standard Euclidean norm for vectors or the magnitude of a real number/complex number/quaternion (which are essentially the Euclidean norm anyway). It would be bizarre if someone used that for an L^p norm.

 

[ice109]

whats the difference between absolute value and norm? in my linear algebra book it is stated that absolute value is the one dimensional case of the norm.

 

[ObsessiveMathsFreak]

whats the difference between absolute value and norm? in my linear algebra book it is stated that absolute value is the one dimensional case of the norm.

There are many different kinds of norms. The one we are most used to is the euclidean, or two norm. 99 times out of a hundred, absolute value refers to the two norm, which for any vector is sqrt(x*x). (I cheat with my notation here). However, it's not the only possible one.

There are taxi cab norms, infinity norms and all other kinds of p-norms. To make a long story short, I think if you simply pick any positive definite matrix A, then (x*Ax)^(k) will do as a norm.

Then there are also semi norms, which can be even more complicated. I've seen |x| used to denote norms, semi-norms and estimates of all kinds.

The basic thing all of these operations have in common is that they take an object, possibly very complicated, i.e. complex number, vector, (quaternion?), function, distribution, whatever, and they return a (positive?) real number that gives you an idea of that objects "size".