All Questions
448 questions
2
votes
1
answer
365
views
Why is an inductive limit of bornological spaces bornological?
Let $(E_\alpha,\tau_\alpha,g_\alpha)$ be a family of bornological (locally convex) topological vector spaces $(E_\alpha,\tau_\alpha)$, where a LCTVS $E$ is said to be bornological if every circled, ...
- 1,247
2
votes
1
answer
1k
views
Norm Closure of a Set
Let $q:\mathbb{R}\rightarrow\mathbb{C}$ be continuous and suppose we look to the set $$A:=\{f\in C_b(\mathbb{R}):\ q\cdot f\in C_b(\mathbb{R})\}$$ as subspace of $C_b(\mathbb{R})$ equipped with the ...
1
vote
0
answers
105
views
The inverse image of a Noetherian topological space
A topological space $X$ is called Noetherian if
closed subsets satisfy the descending chain condition, equivalently,
the open subsets satisfy the ascending chain
condition.
Let $A$ and $B$ be ...
- 11
1
vote
0
answers
82
views
Topology of the algebra $\mathbb{C}\{A\}$ for a LCA group $A$
Let $\mathcal{A}$ be a complex associative Hausdorff topological algebra, and let $A\subset\mathcal{A}$ be a locally compact Abelian (LCA) subgroup (multiplicative). The linear span $\mathbb{C}\{A\}$ ...
- 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 ...
- 1,959
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 ...
- 1,270
11
votes
1
answer
441
views
Example of Banach spaces with non-unique uniform structures
While it is known that compact Hausdorff spaces admit unique uniform structures, it is further shown by Johson and Lindenstrauss's result that Banach spaces are characterized by their uniform ...
- 8,071
3
votes
2
answers
3k
views
Sequentially weak-continuous and continuous functions under the weak-topology of a Banach space
Let $X$ be an infinite dimensional, reflexive and separable real Banach space. Consider a function $f: X \to \mathbb{R}$, and assume $f$ is sequentially continuous with respect to the weak topology, ...
- 349
2
votes
0
answers
192
views
Generalize upper semicontinuous regularization using Borel Hierachy
Let $X$ be a metric space. Suppose a real-valued function $f:X\rightarrow \mathbb{R}$ is upper semicontinuous class $2$ if for all $c \in \mathbb{R},$ its preimage $f^{-1}(-\infty,c)$ is $F_{\sigma}$.
...
- 623
1
vote
1
answer
162
views
Does there exist a class of real-valued upper semicontinuos functions on $X$ such that $\mathcal{F}$ is countable?
Ian Morris quoted the following:
For any upper semi-continuous function $f \colon X \to [-\infty,+\infty)$ defined on a nonempty topological space $X$ there exists a nonempty set $\mathcal{F}\...
- 623
0
votes
0
answers
97
views
Is there any concise sufficient condition for the dual space to have Kadec property?
A normed space $E$ has a
Kadec property if the norm- and weak topologies coincide on the unit sphere of $E$.
Kadec-Klee property if any sequence on the unit sphere, that is weakly convergent is also ...
- 5,529
0
votes
0
answers
84
views
Under what conditions on $\mu^{\beta}$ we have $L_1(\beta X,\mu^{\beta})$ isometrically isomorphic to $L_1(X,\mu)$?
Let $X$ be a locally compact Hausdorff space, $\beta X$ its Stone-Cech compactification and $\Delta: X\to\beta X$ the inclusion map. Given a Borel probability measure $\mu^{\beta}$ over $\beta X$, is ...
- 2,044
9
votes
1
answer
2k
views
When will the supporting hyperplane of a convex set coincide with the tangent?
Due to the supporting hyperplane theorem, a convex set $C$ in a separable topological space has supporting hyperplance at each of its boundary points. The theorem only guarantees its existence, now I ...
- 8,071
7
votes
0
answers
3k
views
What is vague convergence and what does it accomplish?
For convenience, let's say that I have a locally compact Hausdorff space $X$ and am concerned with probability measures on its Borel $\sigma$-algebra $\mathcal{B}(X)$. Natural vector spaces to ...
- 1,124
1
vote
1
answer
134
views
Chain of interior of closed set
It is well known that a topological space with asending chain condition for open subsets is called Noetherian. Is there any characterizations or a nice property for a Hausdorff topological space such ...
- 11
1
vote
1
answer
71
views
Every open convex-valued multimap has global sections?
Let $X$ be a compact Polish space and $Y$ be a separable real Banach space. Assume $U \subseteq X \times Y$ is open, bounded in $Y$-norm, and s.t. for any $x \in X$, $\{y \in Y \mid (x,y) \in U\}$ is ...
- 1,368
1
vote
0
answers
220
views
About the projection on the unit sphere
Let $H$ be a Hilbert Space and let $A\subset H$ be a connected set such that any two elements of $A$ are linearly independent and also $A^{\bot}=\left\{0\right\}$ (this seems to be immaterial). Is ...
- 5,529
31
votes
2
answers
3k
views
Is a normed space which is homeomorphic to a Banach space complete?
I have a normed space $(E,||\cdot||)$ which is homeomorphic (as a topological space) to a Banach space $F$.
Does this imply that $(E,||\cdot||)$ is also a Banach space?
I think I read something ...
- 495
2
votes
1
answer
336
views
Separability of $L^1$ in $L^2$ topology
In the space $L^1(0,1)$ take the topology generated by the $L^2$-balls
$$B^2_r(f)=\{g\in L^1(0,1):\; \|f-g\|_2<r\}.$$
Is $L^1(0,1)$ separable in this topology?
- 23
12
votes
1
answer
1k
views
What is the structure associated to almost-everywhere convergence?
Let $M(X)$ be the vector space (actually it's an algebra) of all equivalent classes of measurable functions $X\to \mathbb{C}$ (where $X$ is a measured space) modulo equality almost-everywhere.
One ...
- 549
2
votes
1
answer
197
views
Explicit description of the closure of a given set
Let $C$ be the subset of $C_b(\mathbb{R})$ given by
$$C:=\{f\in C_b(\mathbb{R}):\ \exists f'\in C_b(\mathbb{R})\}$$
Now I want to take the closure of this set with respect to the supremum norm on $...
11
votes
2
answers
1k
views
How to show that something is not completely metrizable
I have a Polish space $X$ and a subset $A \subset X$.
I know that $A$ is completely metrizable (in its induced topology) if and only if $A$ is a $G_\delta$-set in $X$.
This means: If I want to show ...
- 987
9
votes
0
answers
953
views
Topologies on compactly supported functions
Let M be a (non-compact) smooth manifold and consider the set $C^\infty_c(M)$ of smooth real-valued functions with compact support. We can give this function space several topologies. Here are four:
...
- 27.5k
1
vote
1
answer
118
views
Almost periodic function and closed spaces
We denote $X_{T}$ the vector space of all $T$-periodic function with zero mean in $L^2$ ( we know that $X_{T}$ is spawn by $(e^{2i\pi nt/T})$). Let be $$X=X_{2\pi}+X_{3\pi}.$$
I think that $X_{2\pi}+...
- 75
1
vote
1
answer
134
views
Finding necessary and sufficient topological conditions
$\mathcal G: \mathbb R_+ \to \mathbb R_+$ is a set of strictly increasing continuous functions. If for any $\epsilon>0$,$x\in \mathbb R_+$ and $\alpha\in (0,1)$ there exists $z\leq x$ and $g\in \...
- 21
2
votes
1
answer
265
views
characterization of normality by selection theorem
The Urysohn's extension theorem states that a space $X$ is normal iff every continuous function $f:A \rightarrow \mathbb{R}$, with $A$ a closed subset of $X$, can be extended to a continuous function $...
- 121
4
votes
2
answers
2k
views
What is a generalized limit?
In the proof of Lemma 1.3 in the paper "The ideal structure of a groupoid C* algebra", Journal of Operator Theory 1991 by Jean Renault, I found the notion of a generalized limit of a net without any ...
user1688
0
votes
0
answers
120
views
A topology on the product space of Euclidean space and smooth functions space
I'd like to know if there is a well-known topology on the space $S := \mathbb R \times C^\infty(\mathbb R)$, such that $(x_n, f_n) \to (x, f)$ in $S$ with respect the topology is equivalent to
$$(x_n,...
- 1,399
9
votes
1
answer
777
views
Abstract result on partitions of unity?
A motivation: The classical Stone-Weierstrass theorem says that polynomials are dense among continuous functions (say, on the unit interval), while the abstract Stone-Weierstrass theorem (and also the ...
- 2,479
2
votes
1
answer
358
views
Measurability of integrals with respect to different measures
Let $Y$ be a locally compact Hausdorff topological space (further assumptions like metrizability, separability, etc., may be added if necessary) and let $\mathscr Y$ denote the Borel $\sigma$-algebra ...
- 405
2
votes
0
answers
68
views
Topologies with the same convex closed sets
Let $\tau_1$ and $\tau_2$ be locally convex Hausdorff topologies on vector space $X$ such that $(X,\tau_1)^\ast = (X,\tau_2)^\ast$. It is well known that $(X,\tau_1)$ and $(X,\tau_2)$ have the same ...
- 105
3
votes
0
answers
198
views
Properties of convergence at points of continuity
Let $J$ denote the set of functions $f : [0, \infty) \to \mathbb{R}$ that are right-continuous and have left-hand limits (r.c.l.l.) and such that their points of discontinuity are jumps.
Then $J$ is a ...
- 1,773
4
votes
0
answers
502
views
Every convex sequentially closed set is closed
Let $X$ be a vector space. A vector (not necessarily Hausdorff) topology on $X$ will be called convex sequential if every convex sequentially closed subset of $X$ is closed.
Is there some description ...
- 105
4
votes
1
answer
210
views
Sequentiality of largest vector topology
I know that the largest vector topology on countable dimensional vector space is sequential (i.e. every sequentially closed set is closed). Does it keep for the arbitrary vector space?
In countable ...
- 105
4
votes
1
answer
376
views
Is the topological dual of a Banach space weakly* closed in its algebraic dual?
The question is completely contained in the title :)
I can only add, that it is not difficult to give a counterexample for normed spaces, and also Banach-Steinhaus theorem implies the sequential ...
- 5,529
1
vote
0
answers
305
views
Alternative representation of $C_c(X)$ as inductive limit
CORRECTION: As Simon Henry points out in the comments, there is a problem in the construction: the maps $\varphi_n$ are not necessarily linear.
Under some additional constraints on the space (e.g. $X$ ...
- 1,773
0
votes
1
answer
843
views
$C^{\infty}_{loc}$-convergence - right definition
Let $\Omega \subset \mathbb{R}^{n}$ be some open set. Let $f_{n},f\in C^{\infty}(\Omega)$. My question is: What does the following phrase mean? $f_{n}$ converges to $f$ in $C^{\infty}_{loc}(\Omega)$. ...
- 35
3
votes
1
answer
184
views
Can approximately periodic functions be perturbed to periodic functions on a locally compact group?
Let $G$ be a locally compact group and $H\subset G$ a closed and cocompact subgroup. I wish to consider bounded continuous functions from $G$ to $\mathbb{C}$ that are periodic in the following strong ...
- 1,023
1
vote
0
answers
152
views
Does bounded and closed equal compact for sets of Borel probability measures?
Equip the space of Borel probability measures on a fixed closed subset X of the s-dimensional Euclidean space with the topology induced by weak convergence of probability measures. In this setting, ...
6
votes
2
answers
2k
views
Relation between two different definitions for relative sequential compactness
Building upon this question in Math.SE, I think the following might be rather of interest for MO.
In the literature on measure theory, probability and functional analysis the definition of a subset $...
- 1,773
2
votes
1
answer
244
views
Is the set of entire functions Borel in the space of analytic functions?
$\def\bbR{\mathbb R}\def\bbC{\mathbb C}\def\scrT{\mathscr T}\def\ssp{\kern.4mm}
$More specifically, I ask whether $S$ be a Borel set in the topological space $(\Omega,\scrT)$ in the following ...
- 3,584
1
vote
0
answers
261
views
The closure of a set of closed points
Let $X $ be a compact non-Hausdorff topological space with the following property: for every infinite subset of closed points, say $\{x_i\}_{i \in I}$, there exists $j\in I$ such that $x_j$ is in the ...
- 19
2
votes
1
answer
304
views
Can we Characterise Rings of Continuous Functions?
Suppose $K$ is some nice space, for example $\mathbb R$ or $\mathbb C$. Let $X$ be a set and $C$ a ring of functions $X \to K$. Is there any way to determine, from the algebraic structure of $C$, ...
- 1,955
8
votes
1
answer
360
views
Can we recover a topological space from the collection of Borel probability measures living on it?
Let $(X, \tau)$ be a topological space, and $\mathcal{P}(X, \tau)$ be the Borel probability measures living on $X$. Can we recover $(X, \tau)$ from $\mathcal{P}(X, \tau)$?
user76098
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. ...
- 1,955
-1
votes
1
answer
346
views
An infinite set in a compact space
Let $X$ be a topological space. Is there any characterization for the property that says "for every infinit subset $A$ of $X$ there exists $a\in A$ such that if $f$ be an arbitrary real continuous ...
1
vote
0
answers
178
views
Density of subspace with nonlocal/Wentzell boundary condition
Given the space $F$ defined by:
$$F=\left\{f\in C^2(\mathbb{R}_+^2;\mathbb{R}):f(x,0)=\int_\mathbb{R} f(z,x)g(z)dz, x>0\right\},$$
I want to prove that the subspace $E$ of $F$ defined by $E=\...
- 111
20
votes
2
answers
1k
views
The Gelfand duality for pro-$C^*$-algebras
The Gelfand duality says that
$$X\to C(X)$$
is a contravariant equivalence between the category of compact Hausdorff spaces and continuous maps and the category of commutative unital $C^*$-algebras ...
- 1,344
1
vote
0
answers
62
views
Reference request - Compact embedding of intermediate space
Given two Banach spaces $X_0$ and $X_1$ with norms $\|\cdot\|_0$ and $\|\cdot\|_1$, respectively, such that $X_0\subset X_1$ and $X_0\hookrightarrow X_1$, i.e., $X_0$ is continuous embedded in $X_1$.
...
- 679
2
votes
0
answers
467
views
Reference request: The compactness and compact embedding in Besov Space?
Let $\Omega\subset \mathbb R^N$ be open bounded with smooth boundary. Let $0<s<1$, $1\leq p<\infty$, and $1\leq \theta\leq\infty$. We denote by $B^{s,p,\theta}(\Omega)$ the Besov space. For ...
- 679