I woke up this morning thinking about vector spaces. If that puts you off, this post is not for you. Sorry, Lin. Hello, Stocke.

I had read some Dummit and Foote from a dubiously sourced PDF the night prior, and my mind was spinning. After stumbling to my shelf and getting a copy of Axler, I start messing with an integral that defines an inner product on the vector space of polynomials, $\mathcal{P}(x)$. $$\langle p, q\rangle = \int_0^\infty p(x)q(x) e^{-x} dx,\... Read more

The first “serious” piece I learned for piano was Chopin's Prelude in E minor. It is a languid, lyrical piece, short and quiet with a distinctly funerary air. My edition was entirely unsuited to the emotional quality of the pieces, with a neon cover and a garish, fading blackletter announcing the composers name and the comparative difficulties of the pieces inside.

I took to it quickly. I remember sitting there, timid, at the piano bench, fearing that the downstairs neighb... Read more

Last year, with grueling effort and a little help from my friend Stewie, I managed to complete my school's Calculus I and II courses. Those classes were perhaps the best math classes I've ever taken, seamlessly weaving intuition with elegant formalism ($\delta$-$\varepsilon$ proofs, anyone?). While my summer math efforts have included tours of number theory, combinatorics, the thing that I have spent perhaps the most time on is evaluating integrals.

The process of evaluating integrals enca... Read more