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 […]

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 […]

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

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 […]

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 […]

Intereting Posts

If $f\in C[0,1)$, $\int_{0}^{1}f^{2}(t)dt =\infty$, can one construct $g\in C[0,1)$ so $\int_{0}^{1}g^{2}dt < \infty$, $\int fgdt = \infty$?
Prove $g(x+h) = g(x) + hg'(x) + \frac{1}{2} h^2 g''(x) + o(h^2)$ from definition of limit
Why not just define equivalence relations on objects and morphisms for equivalent categories?
Find the sum $\sum_{n=1}^{50}\frac{1}{n^4+n^2+1}$
Constructing a Möbius strip using a square paper? Is it possible?
Show continuity or uniform continuity of $\phi: (C(;\Bbb R ),||\cdot||_\infty )\to (\Bbb R, |\cdot | )$
Maximum subset sum of $d$-dimensional vectors
A proof for the identity $\sum_{n=r}^{\infty} {n \choose r}^{-1} = \frac{r}{r-1}$
Differential equation of all non horizontal lines?
Showing $(P\to Q)\land (Q\to R)\equiv (P\to R)\land$ {without truth table}
Compact space, continuous dynamical system, stationary point
Combinatorial interpretation for the identity $\sum\limits_i\binom{m}{i}\binom{n}{j-i}=\binom{m+n}{j}$?
Conjugacy Class in Symmetric Group
Proving $\prod_k \sin \pi k / n = n / 2^{n-1}$
Integration with a Bessel function