When would you use the Euclidean norm vs the L2 norm?

lykospirit

Is there a specific reason we integrate from 0 to 1 for the L2 norm? Is it equally valid to integrate from, say, -1 to 0, or from -42 to 783? What about functions that are more "interesting" outside of the interval [0,1] (e.g. f(x) = x if x>2, 0 otherwise)?

adam

In class an example was given of the L2 norm breaking down when there is a discontinuity in the function - zero everywhere except for 1 point would have an L2 norm of 0. What can be can be used in these scenarios?

HelloWorld

I think the issue with discontinuity can be resolved by just requiring the functions to be continuous.

keenan

@cou The Euclidean norm is for ordinary, finite-dimensional vectors (i.e., points in R^n), whereas the L^2 norm is for functions, like the functions described on this slide.

keenan

@lyko Functions on the unit interval are just a specific example on this slide. You can apply the same norm whenever you have a notion of integration of functions (loosely speaking).

keenan

@adam Right, with any definition you have to ask, "what objects does this definition apply to?" In this case, you could do as @HelloWorld suggests and consider only continuous functions. Or, as is common in functional analysis, you could consider the quotient space of functions that differ only bu something in the kernel of the norm, i.e., if two functions differ by something that has zero norm, you consider them to be "the same function." This space of "groups of equivalent functions" still has a vector space structure (adding two "groups" corresponds to adding any two functions from this group and considering the corresponding group), and here there is a unique zero element, comprised of all the functions of zero norm. The L2 norm of any "group" is just the L2 norm of any element of that group.

If you can grok all this, you're doing pretty well at understanding linear algebra! ;-)

When would you use the Euclidean norm vs the L2 norm?

Is there a specific reason we integrate from 0 to 1 for the L2 norm? Is it equally valid to integrate from, say, -1 to 0, or from -42 to 783? What about functions that are more "interesting" outside of the interval [0,1] (e.g. f(x) = x if x>2, 0 otherwise)?

In class an example was given of the L2 norm breaking down when there is a discontinuity in the function - zero everywhere except for 1 point would have an L2 norm of 0. What can be can be used in these scenarios?

I think the issue with discontinuity can be resolved by just requiring the functions to be continuous.

@cou The Euclidean norm is for ordinary, finite-dimensional vectors (i.e., points in R^n), whereas the L^2 norm is for functions, like the functions described on this slide.

@lyko Functions on the unit interval are just a specific example on this slide. You can apply the same norm whenever you have a notion of integration of functions (loosely speaking).

@adam Right, with any definition you have to ask, "what objects does this definition apply to?" In this case, you could do as @HelloWorld suggests and consider only continuous functions. Or, as is common in functional analysis, you could consider the quotient space of functions that differ only bu something in the kernel of the norm, i.e., if two functions differ by something that has zero norm, you consider them to be "the same function." This space of "groups of equivalent functions" still has a vector space structure (adding two "groups" corresponds to adding any two functions from this group and considering the corresponding group), and here there is a unique zero element, comprised of all the functions of zero norm. The L2 norm of any "group" is just the L2 norm of any element of that group.

If you can grok all this, you're doing pretty well at understanding linear algebra! ;-)