Understanding Function Spaces: What Are They?

[coverband]

What is a function space?

thanks

 

[morphism]

In what context?

 

[coverband]

Beginner!

 

[HallsofIvy]

A "function space" is a vector space in which the "vectors" are functions. You often see the simplest examples, "the set of all polynomials of degree less than or equal to 3", say, in Linear Algebra but the phrase normally means infinite dimensional spaces, particularly "L₂", the space of all "square integrable" functions.

 

[coverband]

I thought Euclidean space marked a set of real numbers from X against a set of real numbers from Y and function space marked a set of functions against functions?

 

[slider142]

You're talking about the Cartesian product of X and Y, which provides a (Cartesian) coordinate system for 2-dimensional Euclidean space (a plane). That is to say, each and every point on the plane can be referred to by a unique pair of numbers, one from the set X, and one from the set Y. The Cartesian product of two sets is not itself necessarily representative of Euclidean space (even as a vector space, additional structure must be imposed, such as an inner product or a norm). Otherwise, the pair of numbers may be coordinates within taxicab geometry or hyperbolic geometry, which are both non-Euclidean.
Similarly, a function space is a space of functions (instead of a geometric plane), and you can assign coordinates to the functions in this space (akin to points on the plane) by using some coordinate system. For example, the set of all polynomials is a function space that does not have a finite dimension; each polynomial can be assigned coordinates by the coefficients of the variables. If you allow power series instead of just polynomials, then you get the space of all analytic functions. Some finite-dimensional function spaces are the spaces of polynomials of less than some fixed degree and so forth.
The vector space structure adds the ability to do algebra with these objects.

 

[zhentil]

You take two sets, say X and Y, and a collection of functions mapping X to Y. In many cases, it is required that the collection satisfy certain restrictions.

Some examples.
1) The function f(x)=x is a function space consisting of one "element."
2) The set of all functions f(x) such that f(0)=0 is a function space.
3) The set of all functions from the real numbers to the real numbers is a function space.

If your only exposure to "spaces" is the two-dimensional Euclidean plane, though, I would lay off function spaces until you've seen more math. It's pretty abstract stuff.

 

[skoomafiend]

so question about function spaces.

Is the following a subspace of the set of all functions of the reals? F(R) ={ f | f:R -> R }
The set of all functions f(R) such that | f(1)*f(2) = 0

Would this be considered a subspace? How do I go about testing cases with functions that fit that criteria to see if it is a subspace of f(R)?

Would the zero function be a valid case to test for closure under addition/multiplication? or is that only used to see if there is "zero vector"?

Thanks!

 

[Char. Limit]

Well, for your set to be a subspace, it has to be closed under addition and scalar multiplication. For the first, this means that for f(x) and g(x) in the set, f(x)+g(x) must also be in the set. For the second, for f(x) in the set, then c*f(x) must also be in the set for any scalar c.

A side effect of this second condition is that the zero function, f(x)=0 MUST be in the set for it to be a subspace in R.

Now, the zero condition is easy to prove, and so are the others. So try it, I suppose.

EDIT: As an example of a proof, take f(x)+g(x). Well, for the function to be in the subspace, we know that (f(1)+g(1))(f(2)+g(2)) must be zero. Now just as an example of functions that are in the subspace, let's take f(x)=1-x and g(x)=2-x. We can see that f(1)=0, f(2)=-1, g(1)=1, and g(2)=0. Thus, we need (0+1)(-1+0) to be zero, but it's not, it's -1.

So that's a rather effective counterexample.

 

[skoomafiend]

So for questions like this I am pretty much trying to find counter examples?
I mean the zero vector one would be an easy one and it does satisfy the conditions. But that doesn't mean that the subspace test will not fail under a different test case?

Thanks for your help! I am trying to get a better understanding of function spaces. If you have any recommended reading/sites, please feel free to point me in the right direction.

Thanks again!

 

[Char. Limit]

Note that the set of functions such that f(1)=0 IS a subspace, which is somewhat easy to prove. So if you're looking for something easy to prove that IS a subspace, there's one.

 

[Bacle]

Actually, Morphism's question makes sense;while I have hear of function spaces as vector spaces, when I hear function space I first think of the collection of all functions between two topological spaces as a _topological space_ itself, e.g., one often gives the compact-open topology to the subspace C(X,Y) of X^Y , where X^Y is the collection of all maps between X and Y, and C(X,Y) is the set of continuous functions. But one can also use , e.g., the topology of compact convergence, which I think is the same as the initial topology determined by a collection of seminorms