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Closure of $C([0,1]^2)$ via weak*-topology [closed]

Let $C([0,1]^2)$ denote the set of continuous functions on $[0,1]^2$. Let $L^1([0,1]^2)$ be the set of all Lebesgue integrable functions on $[0,1]^2$. The dual space of $C([0,1]^2)$, denoted by $C^*([...
tom jerry's user avatar
  • 349
0 votes
1 answer
231 views

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 ...
tom jerry's user avatar
  • 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 [...
tom jerry's user avatar
  • 349
-1 votes
1 answer
167 views

Space of distributions on $[0,1]^2$: weakly compact or not?

Let $X_1,X_2$ be distributions on $[0,1]$ and let $X=(X_1,X_2)$ be the joint distribution of $X_1,X_2$. Let $\mathcal{X}$ be the set of all such joint distribution $X$. Question 1: Does $\mathcal{X}$ ...
tom jerry's user avatar
  • 349
2 votes
0 answers
81 views

Extension of a tangent vector field

Let $\Omega$ be an open subset of $S^2$ with $\overline{\Omega} \neq S^2$. Suppose a continuous tangent vector field $G$ is defined on $\partial \Omega$ such that $|G(y)| = 1$ for all $y \in \partial \...
MathLearner's user avatar
0 votes
1 answer
127 views

Continuous extensions of tangent vector fields

Let $\Omega$ be an open subset of $S^2$ with $\bar{\Omega}\neq S^2$. Suppose a continuous tangent vector field $G$ is given on $\partial \Omega$ with $|G(y)|=1$ for all $y\in \partial \Omega$. Does ...
MathLearner's user avatar
0 votes
1 answer
80 views

Continuous modification of tangent vector fields

Let $\Omega$ be an open subset of $S^2$, and assume that there exists a continuous tangent vector field $F(x)$ defined on $\bar{\Omega}\neq S^2$ with $|F(x)|=1$ for all $x\in \bar{\Omega}$. Suppose a ...
MathLearner's user avatar
9 votes
1 answer
428 views

The cardinality of projections of subsets of the Hilbert cube by inner products

I have three related questions. Question 1: Is there a subset $X$ of the Hilbert cube $[0,1]^{\Bbb N}$ of cardinality continuum, such that for each sequence $a\in [0,1]^{\Bbb N}$ with $\sum a_n$ ...
Boaz Tsaban's user avatar
  • 3,104
0 votes
1 answer
192 views

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 $[0,1]^{\Bbb N}$ ...
Boaz Tsaban's user avatar
  • 3,104
1 vote
1 answer
162 views

Is there a uniformly continuous injective image of $(0,1)\setminus\Bbb Q$ in the Cantor space?

It seems too good to be possible, but: Is there a uniformly continuous injective image of $(0,1)\setminus\Bbb Q$ in the Cantor space? Here, the Cantor space $\{0,1\}^{\Bbb N}$ is equipped with the ...
Boaz Tsaban's user avatar
  • 3,104
4 votes
1 answer
177 views

Compact-open Topology for Partial Maps?

I asked the same question on MathStackExchange a month ago and received no answer. I feel that this would be more suitable for MathOverflow. Compact open topology is one of the most common ways of ...
Bumblebee's user avatar
  • 1,093
0 votes
1 answer
92 views

Continuous selectors of a continuous multifunctin on a compact metric space

I am currently working on a continuous selector problem of multifunctions. I am trying to figure out if a continuous multifunction defined on a compact metric space always admit a continuous selector. ...
Saito's user avatar
  • 79
7 votes
0 answers
150 views

The space of analytic associative operations

This question is a follow-up to this old one of mine. Let $\mathcal{A}$ be the set of functions $\star:\mathbb{R}^2\rightarrow\mathbb{R}$ which are associative and $C^\omega$ (real analytic entire) in ...
Noah Schweber's user avatar
2 votes
1 answer
264 views

Is a continuous functional on continuous functions the restriction of a continuous functional on the space of all functions?

As sets, we can consider the space $C(\mathbf{R}^n;\mathbf{R}^k)$ - of all continuous functions from $\mathbf{R}^n$ to $\mathbf{R}^k$ - to be a subset of the product space $(\mathbf{R}^k)^{\mathbf{R}^...
SBK's user avatar
  • 1,179
0 votes
1 answer
253 views

Is the space $L^p_{\text{loc}} (\mathbb R^d)$ separable w.r.t. the norm $\|f\|_{\tilde L^p} := \sup_{x \in \mathbb R^d} \|1_{B(x, 1)} f\|_{L^p}$?

Fix $p \in [1, \infty)$. Let $(L^p (\mathbb R^d), \|\cdot\|_{L^p})$ be the Lesbesgue space of $p$-integrable real-valued functions on $\mathbb R^d$. Let ${\tilde L}^p (\mathbb R^d)$ be the space of ...
Akira's user avatar
  • 835
2 votes
1 answer
281 views

Global control of locally approximating polynomial in Stone-Weierstrass?

Let $X=\mathbb{R}$, and $\mathcal{A}:=\mathbb{R}[x]$ be the subalgebra (of $C(X)$) of univariate polynomials. Given $\varphi\in C_b(X)$ and $K\subset X$ compact, we know from Stone-Weierstrass that $$\...
fsp-b's user avatar
  • 463
1 vote
1 answer
263 views

Does global boundedness ruin Stone-Weierstrass denseness?

Let $X$ be any topological space and denote by $\tau_X$ the topology on $C_b(X;\mathbb{R})$ that is induced by the family of seminorms $(\|\cdot\|_\psi\mid\psi\in B_0(X))$ with $\|f\|_\psi:=\sup_{x\in ...
fsp-b's user avatar
  • 463
0 votes
0 answers
131 views

Cyclic group action and finite invariant set

Let $(X, d)$ be a compact metric space and $G$ a discrete group acting on $X$ such that, for each $g\in G$, the mapping $x\mapsto g\cdot x$ defines a homeomorphism on $X$ Is it true that the ...
Sanae Kochiya's user avatar
0 votes
1 answer
290 views

Tensor product is complete?

Let $(V,\|\cdot\|_V)$ and $(W,\|\cdot\|_W)$ be Banach spaces and let the norm $\|\cdot\|_{V\otimes W}$ on the tensor product space $V\otimes W$ be admissible in the following sense: for $v\in V, w\in ...
Martin Geller's user avatar
4 votes
1 answer
140 views

Whether a functional which preserves maximum for comonotone functions is monotone?

Let $X$ be a compactum (compact Hausdorff space). By $C(X,[0,1])$ we denote the space of continuous functions endowed with the sup-norm We also consider the natural lattice operations $\vee$ and $\...
Taras Radul's user avatar
0 votes
1 answer
243 views

Condition for set of the type $\{(a,b)|a \in A, \ b = f(a)\}$ to have empty interior if $A$ has empty interior [closed]

Let us consider $$\mathcal X = \{(a,b)|a \in A, \ b = f(a)\}, $$ where $A \subset L^1(\mathbb R)$ has empty interior and $f:L^1 \to L^1$ is a bijective map. Does $\mathcal X$ also have empty interior? ...
Riku's user avatar
  • 839
3 votes
2 answers
2k views

Can every real function be approximated with a Riemann-integrable one with any precision required?

Is there some proof that Riemann-integrable functions are dense in the space of all real functions? In a sense that for every real function $f$ and number $\varepsilon>0$, there is Riemann-...
user479568's user avatar
0 votes
0 answers
94 views

Is the space of affine continuous functions a Baire space

Let $\Omega$ be a compact convex set in q linear normed space. Let $A(\Omega)$ be the space of affine continuous real-valued functions. My question is whether the space $A(\Omega)$ is a Baire space? ...
user119197's user avatar
3 votes
1 answer
241 views

$\int_0^1 f(\sin(1/x)) \times g(\cos(1/x)) dx \leq \int_0^1 f(\sin(1/x)) dx \times \int_0^1 g(\cos(1/x))dx? $

I have noticed experimentally that the following question has a positive answer. Is it true that for all even and convex functions $f$, $g$: $$\int_0^1 f(\sin(1/x)) \times g(\cos(1/x)) dx \leq \int_0^...
Dattier's user avatar
  • 4,074
3 votes
1 answer
353 views

Sequential separability on $C_p(X)$

Definition. Let $E$ be a topological space. Suppose that $E$ contains a sequence $\{x_n\}$ such that for every $x\in E$, there exists a subsequence $\{x_{n_k}\}$ of $\{x_n\}$ with $x=\lim x_{n_k}$. ...
ABB's user avatar
  • 4,058
5 votes
1 answer
805 views

Arzelà-Ascoli for $C_b(0,1)$? Or more generally, why is that continuous functions "live most naturally" on compact spaces?

I’m wondering if there is a version of Arzelà-Ascoli for continuous functions on not-necessarily compact metric/Hausdorff spaces $X$, i.e. a characterization of the compact subsets of $C_b(X)$ (under ...
D.R.'s user avatar
  • 831
3 votes
0 answers
187 views

Analogue of Kolmogorov/Arnold superposition for general manifolds?

Previously asked and bountied at MSE with slightly different language: Given a topological space $\mathcal{X}$, let $$\mathsf{Cl_C}(\mathcal{X})=\bigcup_{n\in\mathbb{N}}C(\mathcal{X}^n,\mathcal{X})$$ ...
Noah Schweber's user avatar
5 votes
0 answers
99 views

What is a mild sufficient condition on $X$ such that $C(X, Y)$ is sequential?

Let $X$ be a topological space, $(Y, d)$ a metric space and $C(X, Y)$ the space of continuous maps with the topology of compact convergence. Question: What is a minimal topological condition on $X$ ...
user141240's user avatar
-3 votes
1 answer
315 views

Are the injective functions dense in $C([0,1]^n,\mathbb R^n) $?

Let $n\geq 2$. Are injective functions dense in $C([0,1]^n,\mathbb R^n) $ with the uniform norm?
Dattier's user avatar
  • 4,074
0 votes
1 answer
969 views

Is the pointwise supremum of a continuous function continuous?

Suppose $f(x , y)$ is continuous in both variables. For any $\epsilon > 0$ and some $y_0$, let $h_{\epsilon}(x) = \max_{y^{'}: \| y^{'} - y_0 \| \leq \epsilon} f(x , y^{'})$. It seems to me that $...
Saeed's user avatar
  • 3
4 votes
1 answer
121 views

Condition for existence of a continuous function realizing a partition

Let $\{U_i\}_{i=1}^{I}$ be a non-empty and finite collection of non-empty, disjoint, open, (and obviously bounded) subsets of $[0,1]^n$. Suppose also that $[0,1]^n=\cup_{i =1 }^{ I} \overline{U_i}$. ...
Catologist_who_flies_on_Monday's user avatar
0 votes
1 answer
86 views

Lattice of functions and their minimal separating set upto topological equivalence

There is a very wide series of questions I have been thinking about and I am wondering if there is any literature on this type of structures. Let's start with the set of all functions $F: \mathbb{R} \...
Heraiwa's user avatar
  • 39
0 votes
1 answer
279 views

When does strict inclusion holds for the domain of subdifferential?

Recall that, given an extended real-valued function $f: \mathbb{R}^n \to (-\infty, \infty]$ Its effective domain is, $$\text{dom}(f) = \{x \in \mathbb{R}^n : f(x) < +\infty\}$$ The subdifferential ...
Shamisen Expert's user avatar
2 votes
1 answer
345 views

Function series of normal lower semi-continuous functions

For a real-valued $f$ on a topological space $X$, the upper limit of $f$ at $x\in X$ is defined as follows: $ f^{\ast }\left( x\right) =\inf \left\{ \sup \left\{ f\left( y\right) :y\in U\right\} :U\in ...
Mehmet Onat's user avatar
  • 1,367
12 votes
2 answers
678 views

Non-sequential spaces in the wild

TLDR: What are examples of (function-)spaces that are not sequential? When does this matter? As a simple analyst, I am most happy if I can just work with sequences all the time. In most situations ...
Jan Bohr's user avatar
  • 779
9 votes
1 answer
831 views

Baire category theorem for uncountable unions

Any compact Hausdorff space $X$ is a Baire space: if the set $X$ is a meager set (meaning a countable union of nowhere dense subsets, also known as a set of first category), then $X$ is empty. I am ...
Dmitri Pavlov's user avatar
0 votes
0 answers
147 views

Approximation of Inductive Tensor Product $C(X) \bar{\otimes} C(Y)$

The following question is from Banach Algebra Techniques in Operator Theory written by Ronald G. Douglas. Assume both $X, Y$ are Banach spaces and $X \otimes Y$ is the algebraic tensor product. Let ${...
Sanae Kochiya's user avatar
3 votes
2 answers
676 views

Compact-open limit of continuous functions is continuous?

Let $X$ be a topological space and $Y$ a metric space. A classical result states that compact-open topology on the space $C(X,Y)$ of continuous functions is the same as the topology of uniform ...
user126154's user avatar
3 votes
1 answer
203 views

Regularize continuous functions with bounded variation

Is it true that : $\forall f,g \in C([0,1],\mathbb R), \exists h \in C([0,1],[0,1])$ $f,g,h$ strictly increasing and $h([0,1])=[0,1]$ with $(f \circ h, g\circ h) \in C^{\infty}([0,1],\mathbb R)^2$?
Dattier's user avatar
  • 4,074
-1 votes
1 answer
81 views

Closed on generic set implies closed set whole set [closed]

Assume that $f:\mathbb{R}^{2}\rightarrow \mathbb{R}^{2}$ is a continuous on a set $A$. Let $B \subset A$ be a generic set in $\mathbb{R}^{2}$ i.e, the countable intersection of the open and dense ...
Adam's user avatar
  • 1,043
0 votes
1 answer
223 views

Dense $G_{\delta}$ set with $\sigma$-porous complement is cofinite?

Let $X$ be a separable Banach space and $D\subseteq X$ be a proper, connected, and dense $G_{\delta}$ subset of $X$, $X-D$ is $\sigma$-porous. Then is $X-D$ contained in a finite-dimensional ...
ABIM's user avatar
  • 5,405
4 votes
1 answer
228 views

Haar-null union of dense subsets

Let $\{X_i\}_{i \in \mathbb{R}-\{0\}}$ be a set of subsets of a separable infinite-dimensional Fréchet space $X$ and $I$ be uncountable. Moreover, suppose that (Dense $G_{\delta}$) $X_i$ is a dense ...
MrsHaar's user avatar
  • 63
2 votes
2 answers
316 views

Properties of the topology of sequential convergence $\tau_\text{seq}$

Let $(X,\tau)$ be a Hausdorff space. Denote by $\tau_\text{seq}$ the topology on $X$ whose closed sets are the sequentially $\tau$-closed subsets of $X$. I have read that $\tau_\text{seq}$ has the ...
BigbearZzz's user avatar
  • 1,245
4 votes
1 answer
224 views

Bounded growth of functions vs bounded growth of functions on countable sets

I am wondering if the boundedness of growth can be characterized by sequences. I am not sure if I use the term "growth" correctly, or use the correct tags for this question. Here is what I mean. Let $...
erz's user avatar
  • 5,529
15 votes
3 answers
1k views

Version of Banach-Steinhaus theorem

I am wondering about the following version of the Banach-Steinhaus theorem. Let $A$ be a closed convex subset contained in the unit ball of a Banach space $X$ and consider bounded operators $T_n \in \...
Sascha's user avatar
  • 536
8 votes
0 answers
110 views

Connected component optimization

For an open set $A\subset[0,1]^d$, denote the connected components of $A$ by $cc(A)$. Given a smooth symmetric function $f\colon[-1,1]^d\to\mathbb R$ with $f(0)>0$, I am interested in the ...
Julian's user avatar
  • 623
5 votes
0 answers
349 views

Tietze extension theorem for lower semi continuous functions

On the Tietze extension theorem, if instead of a continuous function "f" we use a lower semi continuous function on a closed subspace of a metric space, is the theorem correct? I mean, can we extend ...
M. Reza. K's user avatar
0 votes
3 answers
554 views

Converting a bounded metric into an unbounded metric

Suppose $d$ is a bounded metric on $X$, i.e. $d(x,y)< K<\infty$ for all $x,y\in X$. Is there a standard way to convert $d$ into another metric $\widetilde{d}$ on $X$ with the property that $\...
JohnA's user avatar
  • 710
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 $F_{\sigma}$. ...
Idonknow's user avatar
  • 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}\...
Idonknow's user avatar
  • 623