Real Analysis - compact and complete
If X is compact prove that C(X,R) is a complete metric space.
Real Analysis - In this space is every closed and bounded set compact?
Let (X,d) be the metric space consisting of m-tuples of real numbers with metric d(x,y)=max{|a_k-b_k|:k=1...m} where x={a_1, a_2,...,a_m} and y={b_1, b_2,...,b_m}. In this space is every closed and bounded set compact?
Real Analysis - Show E is equicontinuous.
Let E be a set of differentiable functions in C[a,b] with uniformly bounded derivatives; i.e., there exists a number M, independent of f in E, such that |f'(x)|<=M for all x in [a,b] and all f in E. Show that E is equicontinuous.
Real Analysis - contraction is continuous
Show that a contraction is continuous.
Real Analysis - Newton's Method show convergence
Newton's Method: Consider the equation f(x)=0 where f is a real-valued function of a real variable. Let x_0 be any initial approximation of the solution and let x_(n+1)=x_n - (f(x_n)/f'(x_n)). Show that if there is a positive number "a" such that for all x in [x_0-a, x_0+a] |(f(x)f''(x))/((f'(x))^2)|<=lambda<1 and |(f(x ...continues
Real Analysis - Banach Fixed Point Theorem
Prove the following generalization of the Banach Fixed Point Theorem: If T is a transformation of a complete metric space X into itself such that the nth iterate, T^n, is a contraction for some positive integer n, then T has a unique fixed-point.
Real Analysis: Show function defines a metric space and the space is complete
Let X be the set of all continuous functions from I_1=[t_0-a_1, t_0+a_1] into the closed ball B[g(t_0);b] is a subset of R_n. Show that for each a>0 the rule d(x,y)=max(|x(t)-y(t)|e^(-a|t-t_0|)) defines a metric on X and that the metric space (X,d) is complete.
Real Analysis - Fredholm equation lipschitz condition
(See attached file for full problem description)
Show that a rule is a metric. See attached file for full problem description.
Real Analysis - Riemann integrals
If f is a function from R to R which is increasing on [a,b], show that f is Riemann integrable on [a,b].
