Linear Spaces

Linear Spaces are like Vector Spaces, but are more general. We will define Linear Spaces, and then use that definition later to define Function Spaces.

If we have a space X, elements in that space f and g, and scalars a and b, the following rules must hold for X to be a linear space:

1. ${\displaystyle f+g\in X}$
2. ${\displaystyle f+g=g+f}$
3. There is a null element φ such that φ + f = f. ${\displaystyle \phi \in X}$
4. ${\displaystyle f\in X,-f\in X}$
5. f + (-f) = φ