I'm trying to understand what one-forms are. The book I'm reading says a one-form is a linear map from a vector to a real number. It uses the gradient as an example but isn't the gradient a map from a function to a vector?
Here's a better way of asking my question: Does a subspace of a space have the same dimensions as the original space? For example, would V3(R) (the space of all vectors of three real numbers) be a subspace of V4(R) (the space of all vectors of four real numbers)? Or is a subspace a restriction...
I have a better way to ask what I want to know. Supposed I use that same definition to define the temperature space of the surface for the Earth using the real numbers as my field F and the set of all points on the surface of the Earth as my T. What would be a subspace of this temperature...
If you define a linear space as the set X of all functions from a nonempty set T into a field F with addition and scalar multiplication defined, would a subspace be the set X restricted to a certain subset of T?
So if we were talking about a Fourier Series, would this be the basis for the Hilbert space:
{0.5, cos x, sin x, cos 2x, sin 2x, cos 3x, sin 3x, cos 4x, sin 4x, ... }
and this be the vector that represents the function?:
{a0, a1, b1, a2, b2, a3, b3, a4, b4, a5, b5, a6, b6, ...}
If...
Yeah, I meant the inner product of a function with itself. Doesn't a space in R^n have to be spanned by n vectors? A Hilbert space would have to be spanned by infinitely many vectors, each with an infinite number of components. That vector you gave is only one vector.
I'm trying to understand Hilbert spaces and I need a little help. I know that it's a vector space of vectors with an infinite number of components, but a finite length. My biggest question is: how is a Hilbert space used to represent a function? Is each component of the vector a point on the...
It seems to me that your time would be better spent reading math books in English than learning another language so that you can read math books in that language. It's not like all the good texts are written in foreign languages or anything. There are plenty of good texts on almost any subject...
Earlier in the math course I'm taking now, we did a block on vector calculus and learned the flux, which is the integral over a surface of the dot product of the vectors and the normal vector in a vector field. Is that similar to what's going on in a Fourier series in any way?
Also, I kind...
I'm having a hard time grasping exactly what a Fourier series is. I know the book definition and that it represents any periodic function as an infinite series. I also know that the a0 term is the average value of the function over one period. I can calculate the terms and everything but I...