What Is An Inner Product Space?

As for the utility of inner product spaces: They're vector spaces where notions like the length of a vector and the angle between two vectors are available. In this way, they generalize $\mathbb R^n$ but preserve some of its additional structure that comes on top of it being a vector space. Familiar friends like Cauchy-Schwarz, the parallelogram rule, and orthogonality all work in inner product spaces.

(Note that there is a more general class of spaces, normed spaces, where notions of length make sense always, but an inner product cannot necessarily be defined.)

The dot product is the standard inner product on $\mathbb R^n$. In general, any symmetric, positive definite matrix will give you an inner product on $\mathbb C^n$. And you can have inner products on infinite dimensional vector spaces, like

$$ \langle \, f, \, g \, \rangle = \int_a^b \ f(x)\overline{g(x)} \, dx$$

for $f, g$ square-integrable functions on $[a,b]$.

This becomes useful, for example, in applications like Fourier series where you want a basis of orthonormal functions for some function space (it's not just the trigonometric functions that work).

An inner product space is a vector space for which the inner product is defined. The inner product is also known as the 'dot product' for 2D or 3D Euclidean space. An arbitrary number of inner products can be defined according to three rules, though most are a lot less intuitive/practical than the Euclidean (dot) product.

Side note:

It may seem slightly esoteric, but as a physicist the obvious application of inner product spaces are Hilbert spaces used in quantum mechanics. The inner product of an eigenfunction with a wavefunction in Hilbert space gives the corresponding eigenvalue.

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).