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Advanced Analysis Spring 2006 Exercises for April 10st: E1: Let X be a topological space. Let x, y ∈ X. Let f : X → C be continuous and assume that f (x) 6= f (y). 1. Show that there exist disjoint open neighborhoods of x and y. 2. Let C(X) be the algebra of continuous functionals on X. Show that A is a subalgebra of C(X) that separates points then X is Hausdorff. 3. What does the above imply for the formulation of the Stone-Weierstraß theorem in the beamer file to Lecture 10? E2: Let H be a Hilbert space. Show that in the universal C ∗ -algebra B(H) (See beamer-file from Lecture 10) the map T → T ∗ is an involution. E3: Let A be a Banach space, and B a normed vector space. Assume that there exists a linear isometry φ : A → B, and that the image of A is dense in B. 1. Show that φ is surjective 2. Show that B is a Banach space. 1