We’re talking about vector spaces, which are quite complicated algebraic structures (i.e., they have two sets, one of which has two binary operations which have to satisfy 10 properties).
More specifically, we’re talking about inner product spaces, which are vector spaces with yet another binary operation (the inner product) that has to satisfy four properties.
In short, this is complicated stuff.
@Mobius, S2K
So given the property of off functions this inner product is not orthogonal? Because the bound is from 0 to 1 and not from -1 to 1 . Can you say why it is or it isn’t true ?
brah, why did you get so obssessed with the (0,1) interval?? That’s something that S2K introduced cause he had to make an assumption since nowhere in your original post did you specify an interval, and orthogonality can only be defined in the context of a particular interval.
You have all your answers here already. The functions are orthogonal on any interval which is symmetrical around zero, (-L,L) where L is an arbitrary number. This true for any combo of real numbers A, K. S2K worked out a formula for the integral evaluated on the (0,1) interval and it’s clearly not zero in general for any real (or integer) A, K. So for specific pairs A, K the functions could be orthogonal on (0,1) but not in general.
It depends on the definition of the inner product space. Which space are we discussing? The one in which functions are defined from 0 to 1, or the one in which functions are defined from −1 to 1. Or some other space?
That’s better.
Cause the 0-1 bound is in the problem. I’m fine doing it without that specific bound but the problem is asking for that bound.Not the negative one to one .