Problem 1. Let H be a Hilbert space and T: H → H be a linear application. Suppose that (Tx, x) > 0 for all x € H. 1. Let (n) be a sequence of H that converges to 0. Suppose that (Tn)n>1 converge to LEH. (a) Show that (L, h) + (Th, h) > 0 for all h € H. (b) Deduce that L = 0. (Hint: replace h by eh for all > 0). 2. Show that the linear application T is continuous.

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