What is a vector space?
In physics students learn to consider a vector to be a quantity that has both a magnitude and a direction.
To mathematicians a vector space is a set that is closed under addition and under multiplication by a scalar. Here closed means that the addition of two vectors yields another vector, and that the multiplication of a vector by a scalar always yields another vector.
The vector quantities familiar to physicists constitute a finite-dimensional vector space.
There are many classes of functions that also have the properties of closure under addition and multiplication by a scalar, and are thus vector spaces in the mathematician's definition. The term linear is synonymous with vector in this usage.
Inner product spaces
A mapping of a space of functions to a space of numbers is called a functional. If there is a functional that is defined for a vector space, then that space is called an inner product space. The classical scalar or dot product of finite dimensional vectors seen in physics is an example of an inner product.
Integration of two functions is the inner product the vector space is a set of functions.