I finally figured out what subspaces are for vectors and matrices, well at least how to do the 3 axions to check anyways, but I'm not sure how to do it for functions since it just saysfand doesn't give a function (if that makes any sense).

F[0,1] = {f | f:[0,1] -> R}

So for example, with a function with a domain of [0, 1] with all real numbers as the output...which of these would be subspaces?

- f(0)f(1) = 0
- f(0) + f(1) = 0
- f(x) = -5f(x) such that x is any real number between 0 and 1

Here is what I figured, I don't know if I'm right or not...

1 is true as it contains a zero vector, and anything multiplied that is zero so it is closed under scalar multiplication, not sure about closed under addition though

3 is true because every number between 0 and 1 is a multiple of negative 5, -5(0) = 0 so it has the zero function, and by adding any two functions with values x between 0 and 1 you still get a number that is a multiple of -5.

I looked over my notes and I still don't quite understand how to prove things by checking if it's closed under addition, but am I right so far for those?