1. Let H be a Hilbert space, Mc H a convex subset, and (x,) a sequence in M such that ||x-d, where d = inf ||x||. Show that (x,,) converges IEM in H. Give an illustrative example in R² or R³. 2. Show that the subset M={y=(n) |Σ n=1} of complex space C" (cf. 3.1-4) is complete and convex. Find the vector of minimum norm in M. 3.1-4 Unitary space C". The space C" defined in 2.2-2 is a Hilbert space with inner product given by (6) 132 (x, y) = §¹ñ+-+§Ñ Inner Product Spaces. Hilbert Spaces In fact, from (6) we obtain the norm defined by ||x||= (₁₁++)²/² = (\ §₁1² + ··· + | £17) 1/2 Here we also see why we have to take complex conjugates ; in (6); this entails (y, x)=(x, y), which is (IP3), so that (x, x) is real.