By definition of a norm, the range of numbers for these functional inner values should be [0, ||f|| * ||g||] right? Does this range allow us to claim a result such as 1/30 is "small"? What about comparing the L2 IP values among different functions with vastly different norms? Is there some kind of normalization we should do to these functions first?
Does this have something to do with convolution? For the similar formula structure they share.
How do we tell what values for the inner product constitute 'lining up'? Just this value alone seems to not be enough since we can let f(x) := 1, g(x) := c for any arbitrarily large constant c, causing the inner product to be arbitrarily large, yet the functions don't line up any more for greater values of c. It seems like it would be better to normalize by dividing the result by the L2 norm of f and g for this purpose.
Does "line-up" mean the similarity in shape of the two functions rather than the values?
I'm having a hard to connecting up the quantity 1/30 with the graph. Does it make sense to try to visualize everything? I can compare two norms e.g. 1/30 and 30, to tell that the former is definitely smaller, but what is the general scale of being large or small.
What practical applications does this have? Aside from maybe how "different" colors are or something similar, its hard for me to imagine an algorithm that would use the L2 norm to calculate an "applicable/useful" value.
Besides inner product of vectors and inner product of functions, is there also inner product of matrices and how is this inner product of matrices useful?
I am confused... is this the inner product of the L2 norm of fx and gx?
oh wait I see it now, if I replace gx with fx, I end up with the L2 norm
What would we do if we wanted to calculate the L2 norm/inner product outside of the unit interval? Would we just change the bounds?