1994 Nov

1994 Nov 16

Instructions. Masters students: Do any 5 problems. Ph.D. students: Do any 6 problems.

1. Let \(E\) be a normed linear space. Show that \(E\) is complete if and only if, whenever \(\sum_1^\infty \|x_n \| < \infty\), then \(\sum_1^\infty x_n\) converges to an \(s\in E\).

2. Let \(f_n\) be a sequence of real continuous functions on a compact Hausdorff space \(X\). Show that if \(f_1 \geq f_2 \geq f_3 \geq \cdots\), and \(f_n(x) \rightarrow 0\) for all \(x\in X\), then \(f_n \rightarrow 0\) uniformly.

3. Let \(f\) be integrable on the real line with respect to Lebesgue measure. Evaluate

   \[\lim_{n\rightarrow \infty} \int_{-\infty}^\infty f(x-n) \left(\frac{x}{1+|x|}\right)\ dx.\]

Justify all steps.