Metric and Hilbert Spaces

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updated: 4 November 2014

Lecture 11: Spaces of functions, uniform convergence

Convergence and boundedness

Let (X,d) be a metric space.

Let AX.

The set A is bounded if A satisfies: there exists M0 such that
if a1,a2A then d(a1,a2)M.

HW: Define the diameter of A, the distance between A and B, and the distance between x and A.

Convergence and boundedness

HW: Let (X,d) be a metric space and let x: >0 X n xn be a sequence in X. Show that if x converges then {x1,x2,x3,} is bounded.

Proof.

Assume x converges.
To show {x1,x2,} is bounded.
We know: There exists xX such that limnxn=x.
To show: There exists M0 such that if i,j>0 then d(xi,xj)M.
Let N>0 be such that if n>0 and nN then d(xn,x)<1.
Let M=2max{1,d(x1,x),,d(xN,x)}.

Notes and References

These are a typed copy of Lecture 11 from a series of handwritten lecture notes for the class Metric and Hilbert Spaces given on August 14, 2014.

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