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8 votes
0 answers
183 views

On "linearly independent" metric spaces

Urysohn's universal metric space $\Bbb U$ satisfies the following surprising property: Whenever $i\colon\Bbb U\to E$ is an isometric embedding into a normed vector space such that $0\not\in i(\Bbb U)$...
Alessandro Codenotti's user avatar
8 votes
0 answers
463 views

When is the sigma-algebra generated by closed convex sets the same as the Borel sigma-algebra

For which topological vector spaces $E$ do we have the equality between the sigma algebra generated by the closed convex subsets of $E$ and the Borel sigma algebra of $E$ ? More precisely, do we have ...
LCO's user avatar
  • 506
6 votes
0 answers
196 views

Logarithm on formal power series continuous?

Denote $V:=\mathbb{R}^d$ and consider the Cartesian product $V^\infty:=\prod_{k=0}^\infty V^{\otimes k}$ together with its canonical projections $\pi_k : V^\infty\rightarrow V^{\otimes k}, (v_0, v_1, \...
fsp-b's user avatar
  • 463
5 votes
0 answers
211 views

A strict directed colimit of Hausdorff locally-convex spaces that is not Hausdorff

We work in the category of locally-convex spaces (morphisms are the continuous linear maps). Let $\Lambda$ be a directed set, for every $\lambda \in \Lambda$ let $V_{\lambda}$ be a locally-convex ...
Jonathan Gleason's user avatar
4 votes
0 answers
108 views

Larger possible chain of closed subspaces in the dual of a Banach space

In this question, is demonstrated that a separable space can have a chain (ordered by inclusion) of closed subspaces with uncountable many subspaces. My question is the following. If $X$ is an ...
Emerick's user avatar
  • 153
4 votes
0 answers
160 views

Pointwise vs. local homotopy equivalences of continuous and smooth complexes of real vector bundles

Let $(E^\bullet,d_E)$ and $(F^\bullet,d_F)$ be two complexes of real vector bundles on a topological manifold $X$, and let $f^\bullet\colon E^\bullet\to F^\bullet$ be a morphism of complexes, i.e. a ...
domenico fiorenza's user avatar
4 votes
0 answers
156 views

Basic calculus on topological fields

Let $K$ be a a topological field (I am mainly interested in the cases when K is either an ordered field or a valued field, e.g. $K = \mathbb Q$ or $ \mathbb Q_p$). 1) Let $f: K^n \to K$ be a ...
Antongiulio Fornasiero's user avatar
3 votes
0 answers
152 views

Can every contractible space be embedded as a convex subset of a vector space?

Given a contractible topological space $X$, is there (or what are some conditions for the existence of) a continuous embedding $\iota:X\hookrightarrow V$ into some topological vector space $V$ such ...
Qfwfq's user avatar
  • 23.3k
3 votes
0 answers
67 views

Non-linear weak*-continuous left inverses

Let $T\colon X\to Y$ be a continuous linear surjection between Frechet spaces. Then it is possible to use Michael's selection theorem to show that there exists a continuous (non-linear) function $g\...
T. Milva's user avatar
3 votes
0 answers
156 views

Topology of the Hamel basis in a TVS

Let $V$ be a complex topological vector space, and let $I$ be a Hamel basis of it. Then as a subset $I\subset V$ acquires an induced topology, becoming a topological space. For a topological space $X$ ...
Bedovlat's user avatar
  • 1,959
3 votes
0 answers
125 views

Commutative discrete cyclic operator groups on topological vector spaces

Let $V$ be a complex Hausdorff separable topological vector space of infinite dimensions. Does there exist a commutative discrete subgroup $A\subset\mathcal{L}(V)$ of continuous operators on $V$ with ...
Bedovlat's user avatar
  • 1,959
3 votes
0 answers
98 views

How Universal is the Topological $\mathbb K$-algebra $C(\Omega, \mathbb K)$?

For $\Omega$ an arbitrary set the family $C(\Omega, \mathbb K)$ of all functions $\Omega \to \mathbb K$ becomes a complete topological $\mathbb K$-algebra under the topology of uniform convergence. ...
Daron's user avatar
  • 1,955
3 votes
0 answers
373 views

Closed Graph Theorem and Spaces Of Continuous Functions

Let $X$ be a (Tychonoff) topological space. Consider $C\left(X\right)$ being a topological vector space of all continuous scalar-valued functions with the compact-open topology. Assume that $Y$ is a ...
erz's user avatar
  • 5,529
2 votes
0 answers
406 views

Complete topological groups in which all subgroups are closed

My previous question has been answered by YCor; so I am asking a new one with a reasonable additional assumption. See the previous question for the background and motivation. General question: does ...
Leonid Positselski's user avatar
2 votes
0 answers
69 views

Invariant compact in division ring

Let $K$ be a discrete valued (with discrete valuation $v$) complete local division ring with ring of valuation $V$. Let $F$ be a compact subset of $V$. Suppose that for all $x\in F$ and for all $y\in ...
joaopa's user avatar
  • 3,998
2 votes
0 answers
459 views

Weak topology on subsets of a normed space

I have few questions about the subsets of a normed space $X$ endowed with the weak topology. Let $E$ be such subset. When is the norm a continuous function on $E$? When is the metric induced by the ...
erz's user avatar
  • 5,529
1 vote
0 answers
111 views

Unique Hausdorff topology on trivial vector bundle?

Question: Is there a Hausdorff topology other than the product topology on $X\times \mathbb{C}^n$, that turns $(X\times \mathbb{C}^n, \mathrm{pr}_1)$ into a vector bundle, where $\mathrm{pr_1}$ ...
PHmath's user avatar
  • 11
1 vote
0 answers
48 views

Neighborhoods of idempotents in topological inverse semigroups

In a topological group, for any neighborhood $U$ of the origin, there is another such neighborhood with the property that $V.V\subseteq U.$ I conjecture a similar property for topological inverse ...
Bumblebee's user avatar
  • 1,093
1 vote
0 answers
81 views

Morphism in commutative square strict?

Let $G,H$ be topological groups and $f:G\rightarrow H$ a continuous group homomorphism. Then $f$ is said to be strict if $G/\mathrm{Ker}(f) \cong \mathrm{Im}(f)$ is an isomorphism of topological ...
KKD's user avatar
  • 473
1 vote
0 answers
120 views

Is the Vietoris topology on compact subsets of $\mathbb R^n$ locally convex?

The title question says it all really. If the question is negative for compact subsets of $\mathbb R^n$, is it affirmative for compact and convex subsets of $\mathbb R^n$? How about for all nonempty ...
aduh's user avatar
  • 869
1 vote
0 answers
53 views

Spaces that are comparable with their compacts

This is an outgrowth of this question. For a (metrizable) space $X$ consider the following increasingly strong properties: (i) For every compact $K\subset X$ there is a map $f:X\to X$ such that $K\...
erz's user avatar
  • 5,529
1 vote
0 answers
122 views

Mackey topology characterising property

Let $V$ be a topological $k$-vector space. Let $V^{\star}$ denote the space of all linear functionals $V \rightarrow k$ and $V' \subset V^{\star}$ the subspace of all continuous linear functionals. ...
user120487's user avatar
0 votes
0 answers
45 views

Critical Growth of Dimension for Dense Cover by Linear Subspaces

Let $X$ be a separable Banach space of dimensional $>2$. When does there exist a sequence positive integers $\{N_n\}_{n \in \mathbb{n}}$ such that For any sequence of distinct finite-dimensional ...
ABIM's user avatar
  • 5,405
0 votes
0 answers
101 views

Can a quotient space of a locally convex space have finer topology that its domain?

The following is related to this post. Let $X=X'$, as sets, and let $T:X \rightarrow X'$ be a surjective map from a countably infinite-dimensional LCS $X$ to itself and equip $X'$ with the final ...
ABIM's user avatar
  • 5,405
0 votes
0 answers
46 views

Show that $\big(s(. |C_n)\big)_n$ is equicontinuous on $X^*$

Let $(X,\|.\|)$ be a separable Banach space with dual $X^*$. $\mathcal{P}_{wkc}(X)$ be the class of nonempty, weakly-compact and convex subsets of $X$. For any $C\in\mathcal{P}_{wkc}(X)$ we define ...
Wer Wer's user avatar