Tagged Questions
4
votes
2answers
198 views
Closed Bounded but not compact Subset of a Normed Vector Space
Consider $\ell^\infty $ the vector space of real bounded sequences endowed with the sup norm, that is
$||x|| = \sup_n |x_n|$ where $x = (x_n)_{n \in \Bbb N}$.
Prove that $B'(0,1) = \{x \in l^\infty ...
0
votes
1answer
91 views
Examples of $T_0, T_1, T_3, T_4$ and Hausdorff spaces
What could be simple examples of $T_0$, $T_1$, $T_3$, $T_4$ and Hausdorff ($T_2$) topological spaces?
1
vote
1answer
21 views
Continuity and openess in quotient space
The setting:
$X$ and $Y$ are topological vector spaces.
$N \subset X$ is a closed subspace.
$T(N)=\{0\}$
$\pi : X \rightarrow X/N$ the quotient map.
$S : X/N \rightarrow Y$ uniquely determined by ...
8
votes
2answers
553 views
If $A$ and $B$ are compact, then so is $A+B$.
This is an exercise in Chapter 1 from Rudin's Functional Analysis.
Prove the following:
Let $X$ be a topological vector space. If $A$ and $B$ are compact subsets of $X$, so is $A+B$.
My guess: ...
3
votes
1answer
131 views
Dual space $E^*$ metrizable iff E has a countable basis
I have trouble proving the following theorem:
If $E$ is a locally convex, Hausdorff topological vector space, then $E^*$ is metrizable if and only if $E$ has an (at most) countable basis.
I've ...
4
votes
2answers
222 views
Example of a topological vector space
I have the following question: give an example of a topological vector space $E$ with subspace $M$ and $N$, such that $E = M \oplus N$ algebraically, but not topologically (so $E \ncong M \sqcup N$). ...
4
votes
2answers
215 views
If you know the convergent sequences, how do you know the open sets?
I have a homework problem which I feel should be simple but is actually surprisingly tricky. This is why I love math sometimes....
Let $X$ be a normed linear space. Suppose $\|\cdot\|_1$ and ...
