Why is this inner product sensible?

Question

I'm trying to understand this statement given by Halmos:

Why does the inner product takes on all possible numerical values? I'm trying to extract this result seeing the cosines formula:

$$\cos(x,y)=\frac{\langle x,y\rangle}{||x||\cdot||y||}$$

Thanks in advance

Answer 1

I've got the book (Finite Dimensional Vector Spaces).

He's talking about why the inner product is a better representaion of the properties of two vectors than the angle between them. In the preceeding paragraphs he describes an inner product in $R^2$, where the angle can range from 0 to $2\pi$.

When you look at $R^1$ then the angle can only be 0 or $\pi$ while the inner product can be any value. So he says that the inner product shows greater sensitivity.

Hope the explanation helps, I'm not sure what conclusion I can draw from Halmos's statement. If it gives you any deep insight into the universe please share it with us.