Undergrad Understanding Subspaces: Definition and Examples | PF Study Guide

[member 428835]

Hi PF!

I want to make sure I understand the notion of a subspace. Our professor gave an example of one: the set of degree n polynomials is a subspace of continuous functions. This is because a) a polynomial is intrinsically a subset of continuous functions and b) summing any polynomials yields another polynomial (closed under addition) and c) any polynomial multiplied by a constant is still a polynomial (closed under scalar multiplication).

If my reasoning is correct, then I think the set of discontinuous functions is NOT a subset of all real functions. To see why, consider $f(x) = 1$ everywhere except non-existent at $x=1$. Then take the function $g(x)=1$ when $x=1$ and non-existent everywhere else. Then the sum is clearly continuous. (How would this change though if I changed non-existent to zero? I know the result is the same, but is $g(x)$ as I have defined it even discontinuous--I realize this is a real analysis question.)

 

[PeroK]

Hi PF!

I want to make sure I understand the notion of a subspace. Our professor gave an example of one: the set of degree n polynomials is a subspace of continuous functions. This is because a) a polynomial is intrinsically a subset of continuous functions and b) summing any polynomials yields another polynomial (closed under addition) and c) any polynomial multiplied by a constant is still a polynomial (closed under scalar multiplication).

If my reasoning is correct, then I think the set of discontinuous functions is NOT a subset of all real functions. To see why, consider $f(x) = 1$ everywhere except non-existent at $x=1$. Then take the function $g(x)=1$ when $x=1$ and non-existent everywhere else. Then the sum is clearly continuous. (How would this change though if I changed non-existent to zero? I know the result is the same, but is $g(x)$ as I have defined it even discontinuous--I realize this is a real analysis question.)

You can't choose functions that are not defined at some point, as functions that have different domains can't be added so they don't form a vector space. The space of all real-valued functions would imply that they all all defined on some fixed domain.

Also, functions that are not defined at a point are not necessarily discontinuous. They may be continuous on their domain. A good example is the function $1/x$, which is a continuous function on its domain.

Instead, you need to think of two genuinely discontinuous functions that can be added to form a contiuous function.

Or, perhaps you could think of a simpler example using a subset of the polynomials that does not form a subspace?

 

[S.G. Janssens]

Our professor gave an example of one: the set of degree n polynomials is a subspace of continuous functions.

I would say: "The set of polynomials of degree at most $n$". (Here you regard the zero polynomial as having degree $-\infty$ or you should stipulate that this set includes the zero polynomial.)

If my reasoning is correct, then I think the set of discontinuous functions is NOT a subset of all real functions.

You mean to say "subspace" instead of "subset"? In addition, I very much second the remarks made by PeroK. They appeared on my screen while I was writing this.

Instead, you need to think of two genuinely discontinuous functions that can be added to form a contiuous function.

Or take a discontinuous function and multiply it by zero.

 

[member 428835]

Thank you both for finessing my logic! I really appreciate both of your input!