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 \(\R\)?