# Vector space

## 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.