One Use For Quotients

If a function almost has some nice property, you can quotient out by that property to make that nice property true.

For example, covariance is a symmetric bilinear form on the space of random variables with finite second moments. But it can be strengthened into an inner product by modding out by difference by a constant. (Two random variables are considered the same if they differ by a constant under this equivalence relation).

This makes covariance positive-definite instead of just positive semi-definite. And now it's an inner product.

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