All Questions
448 questions
1
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1
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71
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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
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1
answer
118
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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
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1
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134
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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
1
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1
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370
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Shrinking $\mathcal{C}_c^{\infty}(M)$ to obtain a first countable space
This is a follow-up to this question.
Let $M$ be manifold (Hausdorff, countable base, finite dimensional, if it simplifies anything embedded in $\mathbb{R}^n$).
I'm interested in the topological ...
- 191
1
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1
answer
179
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Measures idempotent with respect to addition and multiplication.
Does there exist a probability finitely additive measure on $\mathbb N$ which is idempotent with respect to addition and multiplication simultaneously?
It is known (due to Hindman) that there is no ...
- 723
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0
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104
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Commutative Banach $\mathbb{R}$-algebras without complex structure, but with path-connected group of units
For a finite-dimensional commutative (associative, unital) $\mathbb{R}$-algebra $A$, the condition $\pi_0(A^\times) = 1$ (i.e. the group of units of $A$ being path-connected) is equivalent to $A$ ...
- 7,127
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87
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Convergence and sequential compactness for nonlinear operators
I have a family of operators $T_n\colon X \to Y$ where $X,Y$ are Hilbert spaces. These operators are nonlinear.
What kind of notions of convergence does one have for such operators? I'm specifically ...
- 251
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0
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76
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Uniform approximation over compacts using weighted function spaces
I'm interested in approximations over the so-called weighted function spaces.
Let $(X,\tau_X)$ be some completely regular Hausdorff topological space. Additionally, consider some map $\psi: X \to (0,\...
- 161
1
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0
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73
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Is $L^p_\text{loc} (Y)$ dense in $(L^0(Y), \hat \rho)$?
Below we use Bochner measurability and Bochner integral. Let
$(Y, d)$ be a separable metric space,
$\mathcal B$ Borel $\sigma$-algebra of $Y$,
$\nu$ a $\sigma$-finite Borel measure on $Y$,
$(Y, \...
- 657
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0
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99
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Density of Lipschitz functions in Bochner space with bounded support
Let $X$ and $Y$ be separable and reflexive Banach spaces with Schauder bases. Let $\mu$ be a non-zero finite Borel measure on $X$ and let $L^p(X,Y;\mu)$ denote the (Boehner) space of strongly p-...
- 21
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0
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54
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Closed linear span of compact open subsets of a spectral space
Let $X$ be a spectral space and $KO(X)$ be the set of all compact open subsets of $X$. Identify $KO(X)$ with $\{1_D:D\in KO(X)\}$, where $1_D(u) = 1$ if $u\in D$ and $1_D(u) = 0$ if $u\notin D$.
...
- 2,605
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0
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97
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Are Hölder functions between Banach spaces residual in the compact-open topology?
Let $X$ and $Y$ be Banach spaces and let $C(X,Y)$ be the set of continuous functions from $X$ to $Y$ equipped with the topology of uniform convergence on compact sets (i.e. the compact-open topology). ...
- 5,405
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0
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112
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What is the topological characteristic of a separable metric space $X$ such that $|kX\setminus X|=\frak{c}$ for any completion $kX$ of $X$?
What is the topological characteristic of a separable metric space $X$ such that $|kX\setminus X|=\frak{c}$ for any completion $kX$ of $X$?
- 1,101
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540
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Is the set of compact operators closed with the strong topology?
It is well-known that the space of compact operators over Banach spaces is closed within the norm topology.
My question:
Let $X$ be a Banach space.
Considering the strong topology (defined by ...
- 299
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0
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70
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Injectivity of post-composition operator
Let $X$, $Y_1,Y_2$, and $Z$ be separable metric spaces. Let $C(X,Y)$ be the topological space of continuous functions from $X$ to $Y$ equipped with its compact-open topologies. Fix a continuous ...
1
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0
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137
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Relative compactness... but what is the toplogy?
The following Theorem was described in a text I was reading as a compactness result. The proof is probably too advanced for me but I was just wondering with respect to what topology we have ...
- 111
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0
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81
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Examples of spaces which have explicit expression as colimits in $\mathrm{Top}$
$\DeclareMathOperator\Ball{Ball}$Question: What "well-known" spaces can be explicitly written down in the form $\bigcup_k \phi_k C(K_n,\mathbb{R}^m)$; where $K_n$ is a non-empty compact ...
- 5,405
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1
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379
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Creating an inverse system which "stratifies density"
Setting:
Let $X'$ be a dense subset of an infinite-dimensional Fréchet space $X$ and suppose that $(X_n')_{n \in \mathbb{N}}$ is a nested sequence of non-empty subsets of $X'$ satisfying
$$
\bigcup_{n ...
- 5,405
1
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0
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104
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"Global" topologies between compact convergence and uniform convergence
Let $X$ and $Y$ be locally compact (but not compact), second countable, Hausdorff spaces with $Y$ metric. It is easy to see that the topology of compact convergence is weaker than the topology of ...
- 5,405
1
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0
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27
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Approximation of multipliers by multipliers of a smaller set 2
This question is a refinement of my previous question.
Let $X$ be a compact metric space, and let $B$ be a bounded Banach Disk in $C(X)$ such that for every $x\in X$ there is $f\in B$ with $f(x)\ne 0$...
- 5,529
1
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0
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254
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When is the weak topology generated by a family of functions Baire?
Suppose we are given a locally compact space $X$ with $C_b(X)$ denoting the continuous bounded complex or real functions on $X$.
Now, if $A\subset C_b(X)$ is given, I am trying to figure out when the ...
- 173
1
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0
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263
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Norm closure of $C_b^1(\mathbb{R})$
I want to determine what the closure of $C_b^1(\mathbb{R})$, the space of continuous differentiable functions with bounded derivative, with respect to the supremums norm is. I think that $\overline{...
1
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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.
...
- 11
1
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0
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403
views
Dual of $C(X)$ with the compact open topology
Let $X$ be a completely regular space and let $C_k(X)$ be the space of all continuous functions with the compact-open topology. If $X$ is completely metrizable, is the strong dual $C(X)^*$ the strong ...
- 19
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0
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105
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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
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0
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82
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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
1
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1
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162
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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
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0
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220
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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
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0
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305
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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
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0
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152
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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, ...
1
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0
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261
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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
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0
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178
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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
1
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0
answers
62
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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
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0
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233
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Sum-epimorphisms and prod-monomorphisms
Sum-epimorphisms
A longer time ago I have introduced the bi-onto maps for the topological category. Let me formulate here its general categorical definition:
DEFINITION 1 ...
- 7,500
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0
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260
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Generating the sigma algebras on the set of probability measures
I was wondering if somebody could help me see/provide a reference to the following fact: Let $X$ be a metrizable set, $\mathcal{F}$ the corresponding Borel sigma-algebra on $X$, and $\triangle\left(X,\...
- 11
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0
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163
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The category of discontinuous Banach spaces
A banach space is discontinuous if it is isometric to $DC(X)$ for some Hausdorff topological space $X$. ($DC(X)$ is defined here. We denote by $DBan$, the category of all discontinuous ...
- 356
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0
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525
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Separability of the space $C(C[0, 1], \mathbb{R})$
Let $E=C([0, 1])$ be the space of all real-valued continuous functions on $[0, 1]$, equipped with the uniform norm. $C(E)$ stand for the continuous real-valued functions on $E$.
I am wondering that ...
- 199
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0
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331
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Relationship between weak Lp and strong Lq topologies for q<p
Specificaly:
Does convergence in $L^{\frac{1}{2}}$ imply weak $L^2$ convergence?
Having a limit in $L^{\frac{1}{2}}$ topology and a limit in weak $L^2$ topology whether these are always equal? If not,...
- 203
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0
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83
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Topologies on spaces of linear sections
Let $X$ and $Y$ be topological linear spaces which are complete & Hausdorff, and admit dual spaces which separate points. Suppose the topologies are non-separable and non-metrizable.
Let $f : X \...
- 8,512
1
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0
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275
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Regular Borel Measures equivalent definition
Please help me understand how the below definition is equivalent to the standard definition of regularity which says that a measure is regular if for which every measurable set can be approximated ...
- 11
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1
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474
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Hilbert space having all norms (and seminorms) continous.
Suppose I have a Hilbert space $H$ such that every seminorm on $H$ is continuous with respect to the inner-product induced norm. Is $H$ necessarily finite-dimensional? If not, is there an easy ...
- 617
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1
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177
views
$Ax=b$ in a function space
Let
$X$ be compact Hausdorff topological space,
$C(X)$ denote the algebra of complex-valued continuous functions on $X$,
$b\in \mathbb{C}^m$,
$\mathbf{A}\in C(X)^{m\times n}$,
for all $x\in X$, $b\...
- 85
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1
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192
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A continuous injection from the Hilbert cube to the real line?
Continuing an earlier "too good to be true" question that I posted recently, the same holds for the present question:
Is there a continuous injection from the Hilbert cube $[0,1]^{\Bbb N}$ ...
- 3,104
0
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1
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496
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Separability of an algebra is equivalent to separability of its spectrum
Let $A$ be a commutative C*-algebra.
I would like to show that $A$ is separable (i.e. has a countable dense subset) if and only if the spectrum of $A$ (denoted by $\Omega(A)$) is separable.
Notes ...
- 143
0
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1
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249
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About uniform continuity
Is there a definition Df(g) of uniform continuity of g, without using the notion of metric?
Let $(E,d_E)$ and $(F, d_F)$ metrics spaces, $f$ continuous fonction of $E$ to $F$
We must have :
Df$(f)$ ...
- 4,074
0
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1
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536
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About the normability of the space of continuous functions
Let $A$ be a subset of $\mathbb{R}^n$, and denote by $C(A)$ the space of complex-valued continuous functions defined on $A$. We know that if $A$ is compact then we can define a norm on $C(A)$ so that ...
- 1,173
0
votes
1
answer
120
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Breaking up dense subset in non-separable space
Let $X$ be a not necessarily separable (infinite-dimensional) Banach space and $D\subseteq X$ be dense linearly independent subset. Then does there exist a set of infinite-dimensional separable ...
- 5,405
0
votes
2
answers
344
views
subspace topology and strong topology
Suppose $X$ is a locally convex space and $Y$ is a subspace of the strong dual of $X$, is the induced topology on Y equivalent to the strong topology $b(Y,Y')$ on $Y$? If this is not correct, then on ...
- 71
0
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1
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231
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Questions on the compactness of $L_1([0,1]^2)$'s unit sphere
Let $U$ denote the set of functions $f\in L_1([0,1]^2)$ such that $\int f=1$ and $f(x,y)\geq 0: a.e. (x,y)\in [0,1]^2$. Recently in my study I need to study the compactness of $U$. By Riesz's theorem ...
- 349
0
votes
1
answer
101
views
Limit sequence of regular function in $L_1$‘s unit sphere
Let $U$ denote the set of functions $f\in L_1([0,1]^2)$ such that $\int f=1$. For any $f\in U$, we say it is regular if $\int_{x_0\times [0,1]}f=\int_{[0,1]\times y_0}f=1$ for a.e. every $x_0, y_0\in [...
- 349