0
votes
1answer
24 views

Various convergences in the space of bounded operators

Could you please help me to find some classical (counter)examples in functional analysis? Let $X$ and $Y$ be some normed spaces over $\mathbb{C}$. By $\mathcal{B}(X,Y)$ we denote the space of bounded ...
2
votes
1answer
55 views

Counterexample about non Hausdorff topological vector spaces

I have some troubles with Hausdorffness in TVS: Question 1. Is there any topological vector space $X$ which is not Hausdorff? Question 2. Give an explicit example of a topological vector space $X$ ...
0
votes
2answers
156 views

Infinite Dimensional Vector Space: Finite Dim Subspace Closed and Nowhere Dense

Show that any finite-dimensional subspace $(S,\|\cdot\|)$ of an infinite-dimensional normed vector space $(V,\|\cdot\|)$ is closed and nowhere dense. Proof: Let $\{x^{(n)}\}_{n\geq1}$ be a ...
2
votes
0answers
55 views

Is there a non-complete and non-separate metric space?

Is there a (non-trival) non-complete and non-separate metric space? Some notions are here: math.stackexchange.com/questions/182316.
0
votes
2answers
76 views

Example of a sequence of functions

Construct an example of a sequence of functions $(f_n)$ defined on $[0,1]$ such that $f_n$ converges pointwise to $0$ and for every sequence of numbers $(a_n)$ that tends to $\infty$, sequence ...
3
votes
1answer
62 views

Constructing a closed, convex subset of $X^{\ast}$ that is not weakly-* closed

I'm asked to show that if $X$ is a non-reflexive Banach space, there exists (norm) closed and convex subsets of $X^\ast$ that are not $w^{\ast}$-closed. In other words, there's no analogue of Mazur's ...
2
votes
1answer
170 views

Why are the Differential- and multiplication mapping on $C^{\infty}(\Omega)$ continuous?

Let $\Omega\subset\mathbb{R}^n$ be open and $\Omega\neq\varnothing$ and suppose we have the Fréchet topology on $C^{\infty}(\Omega)$ (this can be obtained by the topology construction from out ...
0
votes
1answer
107 views

Equicontinuous sequence of linear maps and a closed subspace

I'm having some difficulty with a homework problem regarding Exercise #14 from Chapter 2 of Rudin's $\textit{Functional Analysis}$. (a) Suppose $X,Y$ are topological vector spaces, $\{f_n\}$ is an ...
3
votes
1answer
108 views

Topological vector space with discrete topology is the zero space

Hello i have a question about topological vector spaces. To remind the definition of such a space: A topological vector space is a pair $(X,\tau)$ with $X$ a vector space and $\tau$ a topology on ...
1
vote
1answer
120 views

Exercise involving topological vector spaces, linear maps, and the quotient map

I'm doing a homework problem out of Rudin's $\textit{Functional Analysis}$ which is basically a proof of which I have completed some of it, but I'm not sure about the rest of it. Without further ado, ...
4
votes
2answers
889 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
120 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
32 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 ...
12
votes
1answer
1k 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
198 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
328 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
253 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 ...