Download Advanced Analysis Spring 2006

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.
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