## 2018-12-29

I continued with linear algebra using Tao’s notes, weeks 7, 8, and 9.

I thought about the following problem a bit, but didn’t get far: if we have an inner product space, we can project a vector onto a subspace to get a best approximation of the vector inside the subspace. But if we start out with a notion of best approximation, can we go from that to an inner product? For instance, the tangent line of a curve through a point is (in a specific sense) the best linear approximation of the curve near that point. Can we now define an inner product (over, say, the polynomials of degree at most $n$) and project an arbitrary polynomial onto the subspace of polynomials of degree at most $1$ so as to recover the tangent line?