  Approximation by polynomials on a triangle is studied in the Sobolev space
$W_2^r$ that consists of functions whose derivatives of up to $r$-th order have
bounded $L^2$ norm. The first part aims at understanding the orthogonal
structure in the Sobolev space on the triangle, which requires explicit
construction of an inner product that involves derivatives and its associated
orthogonal polynomials, so that the projection operators of the corresponding
Fourier orthogonal expansion commute with partial derivatives. The second part
establishes the sharp estimate for the error of polynomial approximation in
$W_2^r$, when $r = 1$ and $r=2$, where the polynomials of approximation are the
partial sums of the Fourier expansions in orthogonal polynomials of the Sobolev
space.
