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Lecture3 - 1 Chapter 3 3.1 Set and Topological...

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1 Chapter 3 3.1 Set and Topological Preliminaries Notation natural numbers integers reals element of for all there exists such that subset of (whether proper or not) union intersection set subtraction implies if and only if Euclidean norm of . Notions from analysis Open ball : Closed ball : Let . is an interior point of if an open ball centered on such that . is open if it consists only of interior points. An arbitrary union of open sets is open. A finite intersection of open sets is open.
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2 Chapter 3 is a neighborhood of if contains an open set containing . A sequence is an ordered list: . Sequence converges to if , an integer such that . This is often abbreviated as . A subsequence of has form: . is a limit point of if a subsequence of that converges to . Example . Consider the sequence . has subsequence that converges to , and subsequence that converges to . and are limit points of . Set is closed if it contains all of its limit points. An arbitrary intersection of closed sets is closed. A finite union of closed sets is closed. Closure of : . Boundary of : . Example . The closed ball is closed. The corresponding open ball is open. . . is bounded if for some . If not, is unbounded . In , compact closed and bounded. Every sequence contained in a compact set has a convergent subsequence. is continuous if such that whenever . means f is continuous on . is uniformly continuous if such that whenever . Note that for a given there must be a that suffices for all . Lemma . A continuous function on a compact set is uniformly continuous.
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3 Chapter 3 Lemma . A continuous function on a compact set attains a maximum and a minimum value on that set. Example . Let is compact but is not. is continuous but not uniformly continuous on . is uniformly continuous on . does not attain a maximum value on but does attain a maximum value on . A sequence of functions converges uniformly to a limit function if such that for all . - , -tube about , with the tube Lemma 3.2 The limit of a uniformly convergent sequence of continuous functions is continuous. Proof . Similar to text (use and ). 3.2 Function Space Preliminaries Let and . The derivative of at a point is given by the Jacobian matrix : . Note that . A function is continuously differentiable on if the elements of are continuous on the open set . We then write . If the k^th partial derivatives of are continuous functions on write . If is continuous with domain and range we can write . Example . Suppose and . Then the chain rule gives . The sup norm of is given by . We will sometimes use the second notation to emphasize that the sup norm is being used.
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4 Chapter 3 Metric Spaces We will often regard functions as points in abstract spaces. These spaces will all be metric spaces. A metric space is a set of elements and a metric . Let . The metric satisfies: and iff (positivity).
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