Forgot some defintions on norms

[Zurtex]

Hi, I've got an exam tomorrow and I completely forgot the difference between the vector norms:

|| \text{v} ||_1 \quad{and} \quad || \text{v} ||_2 \quad{and} \quad || \text{v} ||_\infty

Any help would be greatly appreciated.

 

[matt grime]

I don't think they are standard.

A gues would be,

|v|_n is the n'th roots of the sum nth powers of the absolute values of the components.If they were function spaces then that would make sense, in integrals. Infinity ought to mean the abs value of the largest component, or the sup norm.

 

[HallsofIvy]

Matt! Of course they are standard.

In a finite dimensional space, ||v||\infty= max |v_i|, the largest component, in absolute value, of v.

In an infinite dimensional, function space, ||v||_\infnty= max(|v(x)|) where the max is over whatever compact set v(x) is defined on.

In a finite dimensional space ||v||_1= \Sigma |v_i|.
In an infinite dimensional, function space, ||v||_1= \int |v(x)|dx where the integral is over the set v(x) is defined on.

In a finite dimensioal space ||v||_2= \sqrt{\Sigma (v_i)^2}.
in an infinite dimensional, function space, ||v||_2= \sqrt{\int (v(x))^2 dx}

 

[Zurtex]

Much thanks

Always seem to forget them as most the manipulation needed doesn't require their actual definitions but rather the properties they have. Should help for later today

 

[matt grime]

Matt! Of course they are standard.

you live and you learn.

In a finite dimensional space, ||v||\infty= max |v_i|, the largest component, in absolute value, of v.
In an infinite dimensional, function space, ||v||_\infnty= max(|v(x)|) where the max is over whatever compact set v(x) is defined on.
In a finite dimensional space ||v||_1= \sum |v_i|.
In an infinite dimensional, function space, ||v||_1= \int |v(x)|dx where the integral is over the set v(x) is defined on.
In a finite dimensioal space ||v||_2= \sqrt{\sum (v_i)^2}.
in an infinite dimensional, function space, ||v||_2= \sqrt{\int (v(x))^2 dx}

So they are what i said they ought to be.

I certainly agree that || ||_n is a standard norm on the space of functions on some space, but that information wasn't given in the post, was it? I personally have never seen || ||_n used on ordinary finite dimensional vector spaces. If the question had said norms on Banach spaces then I'd not've been confused. In anycase, topologically the norms on a finite dimensional vector space are all the same.

 

[Zurtex]

Have to I say I am a little confused by what you mean there Matt, this was just for a Topics in Numrical Analysis exam and the vectors were just in R^n.

 

[HallsofIvy]

Ah! Matt's point is that you didn't tell us that! The norms are also defined for infinite dimensional spaces. I gave the definitions in that case for function spaces L_n(C) but they might also be given for the "little l" spaces l_n (infinite sequences of numbers).

Since I now understand that that was the sense in which Matt meant they were "not standard", I must say that I agree with him.

 

[Zurtex]

Oh I see, much apology, I didn't know they were defined for such thing. Bit hard to know there is a generality when you’ve only been taught very specifics.

 

[matt grime]

If it's numerical analysis then there was no chance of me knowing the standard nomenclature, and I can stop worrying that something passed me by. (I know there are interpolations, householder rotations and that's about it, pivots too, perhaps.) I can see why they'd want to use different norms now. For instance, lines of best fit on graphs minimize the | |_2 norm, and not the | |_1 norm, so it starts to make sense.

In general the 'n' norms will make me think of functional analytic beasts, which is why I got confused about the references to (what I presumed were finite dimensional) vector spaces.

 

[D H]

Only one of the L_n norms is particularly conducive to analysis. This is the L_2 norm, whose partial derivative derivatives exist and are linear. The L_2 norm shines so brightly that analysts are drawn to it like moths to a streetlight.

The L_2 norm does not necessarily produce the "best" fit. The L_1 norm produces a "better" fit than does the L_2 norm for error sources with large outliers because the L_1 norm is less sensitive to outliers. In cases where controlling outliers is important, the L_\infty norm produces a "better" fit because this norm is very sensitive to outliers.

By the way, I often am guilty of moth-like behavior. Finding the least-squares fit is easy. Finding some other best fit is much more difficult.