Created: November 06, 2020
Modified: November 06, 2020
Modified: November 06, 2020
transposes are measures
This page is from my personal notes, and has not been specifically reviewed for public consumption. It might be incomplete, wrong, outdated, or stupid. Caveat lector.- According to this reddit post, one of the main takeaways of functional analysis is that the right way to interpret the 'transpose' of a function is as a measure. Apparently this is the intuition for the Riesz-Markov-Kakutani representation theorem.
- I think this makes sense, but I want to try to formalize it.
- In finite-dimensional spaces, the transpose of a vector is a linear functional. We apply that linear functional to a function by taking the dot product of the two vectors. That means summing up their pointwise product.
- In infinite-dimensional space, the transpose of a function should also be a linear functional. And we should apply that functional to a function by integrating their pointwise product.
- So the transpose/functional is 'something you integrate against'. Which makes it a measure. If it normalized to one, it'd be a probability measure.
- Specifically, the Riesz-Markov-Kakutani representation theorem establishes a one-to-one correspondence between continuous linear functionals and Borel measures, for continuous function spaces with compact support or which vanish at infinity (so that integrals are finite).