Prove that all even functions are a linear subspace on the reals. I think you can do this by proving closure of addition and multiplication. Since the sum and scalar product of even functions is still an even function then this is a linear space.

it's interesting if you take the integral of an even function it would just be double the integral of the positive side of the function since it is symmetrical about 0-there must be other properties one could deduce

prove that all rational functions f/g where degree of f<=degree of g are a linear subspace on the reals: