TITLE: What Is An Inner Product Space?
QUESTION [33 upvotes]: As I've understood it, what I've learned is that the dot product is just one of many possible inner product spaces. Can someone explain this concept? When is it useful to define it as something other than the dot product?

REPLY [5 votes]: An inner produce space is a juncture of a vector space with an operation (and is thus an ordered pair consisting of the pair for the vector space $(\mathbb V, \mathbb F)$, and the operation (usually denoted$ \langle,\rangle $). So this would look like $((V,F),\langle,\rangle)$. Obviously there's structure in $\mathbb V$, and $\mathbb F$, and$ (\mathbb V, \mathbb F)$ consistent with a vector space (vector multiplication; scalar multiplication; vector-scalar multiplication; and vector, and scalar addition). The inner product is then a function (with $\mathbb V$ the set of vectors, and $\mathbb F$ the set of scalars):
$$
\langle,\rangle:\mathbb V \to \mathbb F
$$
where the notation $\langle\vec a , \vec b\rangle = c$ means the product of a and b(in $\mathbb V$) reutrns the value C in $\mathbb F$.
there are then a set of properties for the product which must adhere (in order for the product space to have results that are consistent and useful):
I'm going to drop the vector notation above the elements since it should be clear that anything in the product is from the vector space, and any result is a scalar from the field.
$$
\langle a,b\rangle = \overline {\langle b,a\rangle}
$$
which is to say that the complex conjugate of $\langle a,b\rangle$ is $\langle b, a\rangle$. in the real numbers this then becomes simply $\langle a,b\rangle = \langle b,a\rangle$.
The first argument must also be linear, that is (with $x$ a scalar):
$$
\langle xa+y,b\rangle = \langle xa,b\rangle + \langle y,b\rangle
$$
And lastly a relation to some norm on the inner product space:
$$
\langle x,x\rangle \geq 0
$$
That is, a norm can always be defined in terms of the inner product:
$ ||x||$ = $\sqrt{\langle x,x\rangle}$
This norm existing from the definition of scalar multiplication, and the existence of an inner product space proves that there is always a norm in the inner product space (the one defined by the square root of the inner product.
The simplest example other than a dot product is probably the inner product of two functions using integration which returns a value in the scalar filed in which they are defined to be vectors (which has been mentioned previously).