1 Cauchy sequences and \(\R\)

1.1 ​   L04

What is the role of Cauchy sequences with geometric control on differences in the construction of \(\R\) as a quotient space of Cauchy sequences valued in \(\Q\).

1.2 ​   L04

Let \(\Q\subset X \subset \R\) and suppose that any bounded subset of \(X\) has a least upper bound in \(X\). Can you deduce that \(X=\R\)?

1.3 ​   important L04

Think of our construction of \(\R\)? What guarantees that we did not add too much stuff to \(\Q\)? What if we did some other construction and obtained \(\R^2\) instead of \(\R\) where \(\Q\) would be identified with the rationals on the \(x\) axis?

2 Limits in metric spaces

Let \(X\) be a metric space

2.1 ​   L04

Is the limit of \(x_n\in X\) unique?

2.2 ​   L04

Are all Cauchy sequences in \(x_n\in X\) convergent?

2.3 ​   L04

Are all Cauchy sequences in \(x_n\in X\) bounded?

2.4 ​   L04

Are all convergent sequences in \(x_n\in X\) bounded?

2.5 ​   L04

Are all convergent sequences in \(x_n\in X\) Cauchy?

2.6 ​   L04

Give an example of a non-complete metric space that is not a subspace of \(\Q\)

2.7 ​   L04

Give an example of a non-complete metric space that is not a subspace of \(\Q^n\)

3 Metric spaces

3.1 ​   extra food_for_thought L04

How do you "complete" a non-complete metric space?

3.2 ​   L04

What is an isometry?

3.3 ​   L04

Is completeness a metric invariant (is completeness invariant under isometry)?

3.4 ​   L04

Is completeness an invariant under bi-Lipshitz maps?

A map \(f\colon X\to Y\) is Lipschitz if \(\mathrm{dist}\big(f(x);f(x')\big) \leq L \mathrm{dist}\big(x;x'\big)\) for some \(L\gt 0\).

A map \(f\colon X\to Y\) is bi-Lipschitz if it is a Lipschitz bijection with Lipschitz inverse.

3.5 ​   L04

Put a distance on \(\R\) that makes it not complete.

3.6 ​   L04

A homeomorphism \(X\to Y\) is a bijection that is continuous with continuous inverse.

  • Give an examle of a homeomorphism that is not an isometry.
  • Give an examle of a homeomorphism that is not an bi-Lipschitz.

3.7 ​   L04

Is completeness a topological invariant (is completeness invariant under homeomorphism)?

4 Banach spaces

4.1 ​   L04

Are all finite dimentional vector spaces over \(\R\) complete?

4.2 ​   L04

Can you come up with a non-complete finite dimensional vector space?

4.3 ​   L04

Is \(\mathrm{Lin}(\R^{m};\R^{n})\) a vector space?

4.4 ​   L04

Is \(\mathrm{Lin}(\R^{m};\R^{n})\) complete? (Did I forget to specify something?)

4.5 ​   L04

  • Is \(\mathrm{Cauchy}(\Q)\) a vector space over \(\R\)?
  • Is \(\mathrm{Cauchy}(\Q)\) a vector space over \(\Q\)?
  • Is \(\mathrm{Cauchy}(\R)\) a vector space over \(\R\)?