We generalize the concept of a dot product in a vector space, ‘upgrading’ it with a new definition which will allow us to introduce geometric concepts such as:
- angle
- distance
NOTE: For this section, we use $\langle \mathbf x, \mathbf x \rangle$ to denote inner products instead of $\mathbf x \cdot \mathbf x$
An even more general characterization of the ‘ruler’ that inner product spaces introduced. We extend this to any linear functional that maps a vector space to its field.