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is continuous on X. (b) The inverse image of every closed set in Y is closed in X. (c) The inverse image of every open set in Y is open in X. (d) For every...

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13-11.1. Let H be a Hubert space. Suppose A is an operator on H. Recalled that a complex number A is called an eigenvalue of A if there is a non-zero vector...

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contains a point in M and also a point not in M. The set of all boundary points of M is called the boundary of M and is denoted by dM.

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Then the support of / is defined to be the closure of the set {x € X : f(x) ^ 0}. It is denoted by supp(f). Let A be a closed subset of X and

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be a metric space. A sequence {x n } in X is said to be Cauchy if for every e > 0, there is an integer p such that for all m,

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is said to be uniformly continuous if for every e > 0, there is 6 > 0 such that for all

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0. • 1-8.5. Theorem Let A, B be disjoint closed subsets of a metric space X. Then there is a continuous function / : X — > [0, 1] such that

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is a continuous bijection which is not a homeomorphism. 2-7.8. Exercise A subset of a metric space is said to be relatively compact if its closure is compact....

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The set H is said to be equicontinuous on X if it is equicontinuous at every point of X. For every x € X , write