# Notes

Often when learning something new I try to put it into my own words in order to understand it better. Here are some random examples. Feel free to have a look, however since those are not proofread treat them carefully and please let me know any small and big mistakes you find!

**Expressivity of Neural Networks:** Coming soon…

**Reproducing kernel Hilbert spaces:** Coming soon…

**ODEs in Fréchet spaces:** Other than in Banach spaces ODEs are not well posed for Lipschitz continuous vector fields. We discuss that the Picard-Lindelöf theorem breaks down because Bochner’s inequality doe not hold in metric vector spaces. In fact we see that the inequality is equivalent to the statement that the space is in fact a Banach space. The counterexample is adapted from here.

**Parameter estimation for discrete determinantal processes:** Coming soon…

**Random fields:** I compiled those notes in the preparation for an oral exam at the University of Warwick.

**Measures as dual spaces:** Finitely additive measures naturally induce linear forms on measurable functions. Even more, they form exactly the dual space of bounded measurable functions. Further, the $\sigma$-additivity of a measure is equivalent to the theorems of monotone and dominated convergence.

**Characterisation of Hilbert spaces:** One can explicitely characterise Hilbert spaces and show that they are ismetrically isomorphic to $L^2$ spaces with respect to the counting measure. Using this approach one can show the representation theorem of Riesz just like in linear algebra; see also those lecture notes by Wolfgang Soergel.

**Set topology for functional analysis:** This is a little list of elementary set topology that where useful to me when solving exercise sheets in functional analysis.