What does inner product actually mean?

Question

What does inner product actually mean? So far most of the cases that I encounter seems to suggest that dot product is the only useful inner product. I mean most of the things that we discuss about defines inner product as dot product. For example, when we talk about norm or length of a vector. If that is the case, why do we need inner product? I don't really get the purpose of defining such thing.

Answer 1

Inner product is a generalization for dot product on a space $X$. And with this generalization, we can define the angle between two elements in $X$ and we can talk about perpendicular, projection of one element on the other element's direction, minimum distance and so on.

For example, we can define an inner product on $C[0,1]$ by $<f,g>=\int_{0}^1 f(x)g(x)\, dx$.

Answer 2

The inner product is not necessarily the best way to think of it; I would argue that orthogonality is the more useful concept. It's nice to know when things are perpendicular, and the inner product provides us with a way of determining that in contexts more general than, say, $\mathbb{R}^N$.

Why is it useful? Well, it's nice to be able to decompose vector spaces (even infinite dimensional ones!) into collections of mutually orthogonal subspaces.

For example, given a subspace $V \subset W$ of some vector space, when do you know that you can write $W = V \oplus V'$ in a nice way? Well, if there is an inner product on $W$ then there is a natural way to do so; just define $V' = V^\perp$. Without that there is not necessarily a natural choice of complement.

Answer 3

Inner product tells you how much of one vector is pointing in the direction of another one. If e is a unit vector then $<f, e>$ is the component of f in the direction of e and the vector component of f in the direction e is $<f, e>e$. The vectors f and e are orthogonal when $<f, e> = 0$, in which case f has zero component in the direction e.
In 2 dimensions and with dot product this is familiar. We can write any vector v as $<v, i>i + <v, j>j$ where i and j are unit vectors in the direction of the x and y axes.

More interestingly, consider the space of real square integrable continuous functions on the interval [0, 1] and inner product $<f, g>=\int_{0}^{1}f(x)g(x)dx$. With respect to this inner product the functions $u_n=sin(2n\pi x)$ and $v_n=cos(2n\pi x)$ are orthonormal for each n, just like the usual basis vectors in n dimensions. A function (vector) in our space can now be written as (modulo some technicalities) $$\sum_{0}^{\infty}(<f,u_n>)sin(2n\pi x)+<f,v_n>cos(2n\pi x))$$

the real fourier series for f. That is one use of the inner product - it lets you visualize function spaces geometrically.