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?