Metric Spaces: Complete Metric Spaces

Cauchy criterion: for all \(\epsilon > 0\) there exists \(N_\epsilon\) such that \(\rho(x_n, x_{n^\prime}) < \epsilon\) for all \(n, n^\prime > N_\epsilon\). A sequence satisfying this criterion is called a Cauchy sequence or fundamental.

Theorem 1

Every convergent sequence \(\{x_n\}\) is fundamental.

Complete if every Cauchy sequence in \(R\) converges to an element of \(R\). Incomplete otherwise.

Theorem 2 (Nested sphere theorem) Complete if and only if every nested sequence \[\{S_n\} = \{S_\infty [x_n, r_n]\}\] of closed spheres such that \(r_n \rightarrow 0\) as \(n \rightarrow \infty\) has a nonempty intersection \(\cap_{n=1}^\infty S_n\).

Theorem 3 (Baire) A complete metric space cannot be represented as the union of a countable number of nowhere dense sets.

\(R^*\) completion of \(R\) if \(R \subseteq R^*\) and \([R] = R^*\).

Theorem 4 Every metric space \(R\) has a completion. This completion is unique to within an isometric mapping carrying every point \(x \in R\) into itself.

Problems

  1. Prove that the limit \(f(t)\) of a uniformly convergent sequence of functions \(\{f_n(t)\}\) continuous on \([a, b]\) is itself a function continuous on \([a, b]\).

    Hint. Clearly \[|f(t) - f(t_0)| \leq |f(t) - f_n(t)| + |f_n(t) - f_n(t_0)| + |f_n(t_0) - f(t_0)|\] where \(t, t_0 \in [a, b]\). Use the uniform convergence to make the sum of the first and third terms on the right small for sufficiently large \(n\). Then use the continuity of \(f_n(t)\) to make the second term small for \(t\) sufficiently close to \(t_0\).

    Do as the hint says.

  2. Prove that the space \(m\) is complete.

    \(\rho(x_n, x_{n^\prime}) = \sup_k (x_{n_k} - x_{{n^\prime}_k}) < \epsilon\)

  3. Prove that if \(R\) is ocmplete, then the intersection \(\cap_{n = 1}^\infty S_n\) riguring in Theorem 2 consists of a single point.

    \(R\) complete, so limit in \(R\) by definition. And by Theorem 2, intersection nonempty.

  4. The diameter of a subset \(A\) of a metric space \(R\) is \[d(A) = \sup_{x, y \in A} \rho(x, y)\] Suppose \(R\) complete, and let \(\{A_n\}\) be a sequence of nested of closed subsets of \(R\). Suppose further that \(\lim_{d \rightarrow \infty} d(A_n) = 0\). Prove that the intersection \(\cap_{n = 1}^\infty A_n\) is nonempty.

    Same as the nested sphere theorem. Also use definition of complete.

  5. A subset \(A\) of a metric space \(R\) is said to be bounded if its diameter \(d(A)\) is finite. Prove that the union of a finite number of bounded sets is bounded.

    There’ll always still be a lowest and hightest number.

  6. Give an example of a complete metric space \(R\) and a nested sequence \(\{A_n\}\) of closed subsets of \(R\) such that \(\cap_{n=1}^\infty A_n = \emptyset\). Reconcile this example with Problem 4.

    Take all of the subsets to be the empty set. The diameter limit no longer holds.

  7. Prove that a subspace of a complete metric space \(R\) is complete if and only if it is closed.

    Both ways by definitions.

  8. Prove that the real line equipped with the distance \[\rho(x, y) = |\arctan x - \arctan y|\] is an incomplete metric space.

    \(\frac{\pi}{2}\).

  9. Give anxample of a complete metric space homeomorphic to an incomplete metric space.

    \(f(x) = \frac{2}{\pi} \arctan x\)

  10. Construct the real number system.

    Hint. If \(\{x_n\}\) and \(\{y_n\}\) are Cauchy sequences of rational numbers serving as ‘representatives’ of real numbers \(x^*\) and \(y^*\), respectively, define \(x^* + y^*\) as the real number with representative \(\{x_n + y_n\}\).

    Also need to define the products, and verify that the usual axioms of arithmetic are satisfied.