What's the purpose of defining the inner product?

Question

Is the inner product simply a definition that conveniently helps to give the notion of the norm and distance between points or vectors in a real vector space? I understand it gives the Euclidean space a sort of 'structure', but it seems like a quite artificially constructed definition. It seems to make calculations easy, but my question is, what would be the purpose of the inner product in and of itself?

Answer 1

Inner products neatly encapsulate and extend Euclidean-ness. Whenever an inner product space $n$-dimensional, it is structurally identical to $\mathbb{R}^n$ or $\mathbb{C}^n$, depending on whether it's a real or complex space.

Although I'd agree that the definition itself is somewhat artificial, it is easy to apply, while giving us an enormous amount of structure. It permits for geometry to be done in more abstract spaces, and sometimes lends itself to some elegant algebraic proofs of purely geometric facts (e.g. orthogonality of diagonals of a rhombus becomes essentially a difference of two squares argument).

The benefits go further into infinite dimensions. Often, inner product spaces are required to be complete (in the sense of convergent Cauchy sequences), giving us Hilbert Spaces. In such spaces, we can consider things like Fourier Series: an easy yet powerful and efficient tool for approximation of functions, as well as other tricky infinite-dimensional problems.

Correspondingly, you also get tools like the Fourier Transform, which is invaluable in many applied settings, particularly notably, in the manipulation of sound. For example, if you take a soundwave and compress it, the sound plays faster and more high-pitched. If you want to make a soundwave simply play faster or play with higher pitch (but not both), then you need to use the Fourier Transform.

I guess a more natural definition would be a the direct definition of an angle. However, in practice, a sensible definition would end up equivalent, and likely end up a lot harder to put into practice, or perform calculations with. Inner products are defined to be powerful and elegant.

Answer 2

Thanks to the inner product, we have the angle $\theta$ between two non-zero vectors $u$ and $v$; it's the number $\theta\in[0,\pi]$ such that$$\cos\theta=\frac{\langle u,v\rangle}{\|u\|.\|v\|}.$$

Answer 3

As you know many concepts such as projection of a vector on another vector, orthogonality, finding the angle between two vectors, and as you mentioned the concepts of norm and distance are easily explained by dot product.

In Numerical Analysis, most quaratures are simply a dot product of two vectors, such as $$w_1 f(x_1) + w_2 f(x_2) +...+ w_n f(x_n)$$

You may downgrade the dot product because it is easy to compute but is indeed a virtue of dot product not a failure.

Compare it with the cross product and you will like it.