Let E be an arc on the unit circle and let L 2 ( E ) be the space of all square integrable functions on E . Using the Banach–Steinhaus Theorem and the weak* compactness of the unit ball in the Hardy space, we study the L 2 approximation of functions in L 2 ( E ) by polynomials. In particular, we will investigate the size of the L 2 norms of the approximating polynomials in the complementary arc E of E . The key theme of this work is to highlight the fact that the benefit of achieving good approximation for a function over the arc E by polynomials is more than offset by the large norms of such approximating polynomials on the complementary arc E .