Articles of complete spaces

When is the universal cover of a Riemannian manifold complete?

Let $(M,g)$ be a connected Riemannian manifold which admits a universal cover $(\tilde{M}, \tilde{g})$, where $\tilde{g}$ is the Riemannian metric such that the covering is a Riemannian covering. I want to know under what conditions the universal cover $\tilde{M}$ is complete. The reason for this questions is that I want to know under what conditions […]

Fixed points of contractions in metric spaces

How do I prove that all contractions on a complete, non-empty metric space has exactly one fixed point? What I know: I know that all contractions are continuous and that completeness of $A$ means that all Cauchy sequences converge in $A$, but what can I do with these facts? And I think I also have […]

Can the real vector space of all real sequences be normed so that it is complete ?

Let $X$ be the vector space of all real sequences . Does there exist a norm on $X$ which makes it complete ?

Showing that if a subset of a complete metric space is closed, it is also complete

Let $(X, d(x,y))$ be a complete metric space. Prove that if $A\subseteq X$ is a closed set, then $A$ is also complete. My attempt: I tried to prove that every Cauchy sequence $(b_n)$ of points of $A$ converges to a point $b\in A$. However could not figure out the exit way. Maybe I am on […]

Show that $l^2$ is a Hilbert space

Let $l^2$ be the space of square summable sequences with the inner product $\langle x,y\rangle=\sum_\limits{i=1}^\infty x_iy_i$. (a) show that $l^2$ is H Hilbert space. To show that it’s a Hilbert space I need to show that the space is complete. For that I need to construct a Cauchy sequence and show it converges with respect […]

The completion of a separable inner product space is a separable Hilbert space

Let $X$ be a separable inner product space and $\bar X$ its completion. Then I want to check that $\bar X$ is a separable Hilbert space. But I am stuck here, because how do you prove that $\bar X$ has a inner product induced by the one of $X$?, and how do you construct a […]