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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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0 1-8.5. Theorem Let A, B be disjoint closed subsets of a metric space X. Then there is a continuous function f: X —‘ [0, 1] such that f(A) = 0 and...

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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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13-3.4. Theorem Let B be an orthonormal set in a Hilbert space H. Then the following statements are equivalent. ( a) B is an orthonormal basis, ie a maximal...

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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

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1-7.3. Exercise Let M = (1,2] U {¿ : n > 1) be the union of a semiinterval and a sequence in the real line. How many new sets can you obtain by constructing...

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The trace of a square matrix is the sum of its diagonal entries.

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imply x ¿ z, transitive. A set together with a partial order is called a partially ordered set, or a

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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