Skip to main content

All Questions

Filter by
Sorted by
Tagged with
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
0 votes
1 answer
254 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
  • 825
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
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
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
0 votes
2 answers
323 views

Is the restriction map $C^1\ni f\mapsto\left.f\right|_K$ a continuous map?

Let $E$ be a $\mathbb R$-Banach space, $\Theta\subseteq C^{0,\:1}(E,E)$ be a $\mathbb R$-Banach space and $\iota$ be a continuous embedding of $\Theta$ into $C^1(E,E)$. I would like to show that, ...
0xbadf00d's user avatar
  • 167
0 votes
1 answer
81 views

If $\tau_1\subset \tau_2$ and $X^*$ is separable for $\tau_1$ then $X^*$ is separable for $\tau_2$?

Let $X$ be a Banach space the associated dual space is denoted by $X^*$. Take $\tau_1$ and $\tau_2$ two topologies in $X^*$ compatible with the duality $(X^*,X)$, such that $\tau_1\subset \tau_2$. ...
Karim KHAN's user avatar
0 votes
1 answer
114 views

$ \overline{(A-A)}\cap\overline{B}(0,r)\text{ is weakly compact, }\forall r>0 $?

Let $X$ be a separable Banach space and $A$ is a subset of $X$ such that $$ A\cap\overline{B}(0,r) \text{ is weakly compact, } \forall r>0. $$ Can we say that : $$ \overline{(A-A)}\cap\overline{...
kaka Hae's user avatar
  • 117
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 $C^{\infty}_{loc}(\Omega)$. ...
Ben's user avatar
  • 35
0 votes
1 answer
208 views

The pth power of a distance function is twice continuously differentiable, for $p>2$?

Suppose $\mathcal{O}$ is an open convex connected strict subset in $\mathbb{R}^n$ and define $\beta(x)=dist(x, \mathcal{O})$, for each $x\in\mathbb{R}^n$. Is $\beta^p$, $p>2$ a twice continuously ...
Xi LIN's user avatar
  • 9
0 votes
1 answer
319 views

Hilbert space automorphisms realized as induced by transformations of some base-spaces

Following question may be soft. Fix abstract hilbert space H and consider any automorphism A in banach-spaces sence (i.e. no conditions on metric). Call A is realizable if exist measure space $(X,\mu)$...
Bad English's user avatar
0 votes
1 answer
291 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
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
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
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
81 views

Ultrabornological representation for the space of uniformly continuous functions?

Let $\{\omega_i\}_{i\in I}$ be a non-empty set of increasing (not necessarily strictly) continuous functions preserving $0$. Then, for each $i \in I$ define the space $$ C_{\omega_i}(\mathbb{R}^n,\...
ABIM's user avatar
  • 5,405
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
0 votes
2 answers
796 views

Extending Continuous Sublinear maps on dense subsets of a Banach space

Suppose X' and Y are Banach spaces and X is a linear subspace dense in X'. Let T be a continuous map of X to Y satisfying: (1) ||T(x+y)|| is less than or equal to ||T(x)||+||T(y)||. Please prove ...
Jeffrey's user avatar
  • 11
0 votes
0 answers
72 views

Sequential compactness via Arzela-Ascoli theorem for uniform state spaces

Let $X$ be a uniform topological space and $C([0,1],X)$ the space of continuous functions from [0,1] to $X$. Assume that for subsets of $X$ sequential compactness and compactness are equivalent. Let $(...
PDEprobabilist's user avatar
0 votes
0 answers
97 views

Generator of an analytic semigroup

Perhaps I have a naive question. My question is as follows: When we consider a Cauchy proposition of the following form: $$ \begin{cases} x'(t)= -Ax(t)+ F(t,x(t)) &\text{for}\ t> 0 \\ x(0)=...
Mathlover's user avatar
0 votes
0 answers
98 views

Does weak $L^2$ approximation implies $L^2$ approximation under a condition similar to convexity?

(Cross posted from Math StackExchange: Does weak $L^2$ approximation implies $L^2$ approximation under a condition similar to convexity?) Assume $(\Omega, \mu)$ is a probability space. Consider a ...
David Gao's user avatar
  • 2,830
0 votes
0 answers
77 views

Completeness of a normed space

We consider the set $\mathcal{PC}([-r,0],X)$ $$\mathcal{PC}([-r,0],X):=\{\varphi:[-r, 0] \rightarrow X: \varphi \text{ is continuous everywhere except for a finite number of points } t_* \text{ ...
Mathlover's user avatar
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
0 answers
98 views

Reference request: subspace-based generalisation of weak* convergence

Let $V$ be a normed space and $(V_j)_{j\in [0,1]}$ be a family of linear subspaces of $V$ with $V_1$ non-trivial and such that $V_1\subsetneq V_j\subseteq V_i$ whenever $i\leq j$. We write $W:=V'$ for ...
fsp-b's user avatar
  • 463
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
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
107 views

Open subset of compact-open topology

Let $E$ be a Banach space, $X$ a locally-compact metric space, equip $C(X,E)$ with the compact-open topology. Let $F:E \rightarrow E$ and consider the induced map $F_{\star}(f):=F \circ f$ on $C(X,E)$...
ABIM's user avatar
  • 5,405
0 votes
0 answers
113 views

Viewing limit as a map

Question: Let $X$ be a set of functions from $\mathbb{R}$ to itself. Consider the subset $X_0$ of the sequences $(f_n)_{n=1}^{\infty}\in X^{\mathbb{N}}$ for which $$ f_{\infty}(x) = \lim\limits_{n \...
ABIM's user avatar
  • 5,405
0 votes
0 answers
102 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
60 views

The scalar convergence in $\mathcal{C}(X)$ is topologizable?

Let $(X,\|.\|)$ be a separable Banach space and $\mathcal{C}(X)$ be the collection of all nonempty, closed and convex subsets of $X$. For any $C$ in $\mathcal{C}(X)$ we set $$ s(x^*, C) := \sup_{x\in ...
Made's user avatar
  • 115
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
0 votes
0 answers
66 views

Coincidence Topologies for $L^p$ spaces

If $X$ and $Y$ are compact metric spaces then it is well-known that the compact-open topology on $C(X,Y)$ coincides with the topology of uniform convergence on compacts. Therefore, the latter is ...
ABIM's user avatar
  • 5,405
0 votes
0 answers
142 views

Mackey topology

Recall that for a Hausdorff locally convex space $X$ the Mackey topology $\tau (X^*,X)$ is the topology in its topological dual $X^*$ of uniform convergence on all weakly compact absolutely convex ...
Wer Wer'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
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 ...
erz's user avatar
  • 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 ...
Leandro's user avatar
  • 2,044
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,...
kenneth's user avatar
  • 1,399
0 votes
0 answers
114 views

Recontruction of the weak topolgy from the scalar product on a subset of a Hilbert Space

Let $M$ be a set a let $K:M\times M\to\mathbb{C}$ be a positive definite kernel. By a version of Moore-Aronszajn Theorem, there is a unique (up to the unitary euivalence) Hilbert Space $X$, and a map $...
erz's user avatar
  • 5,529
0 votes
0 answers
153 views

extension of function in an abstract metric space

my question is the following.(Maybe my title is not quite proper for this question): Let $(E,d)$ be a Polish space (or a separable metric space), let $\xi: E\to R_+$ be a Lipschitz function. Now set $...
CodeGolf's user avatar
  • 1,835
-1 votes
2 answers
409 views

$X$ is Polish and $N$ is countable. Is $N^X$ Polish? [closed]

$X$ is a separable, completely metrizable topological space equipped with its sigma algebra of Borel sets. $N$ is a countable space. $X^N$ is the collection of all mappings from $N$ to $X$. It is ...
High GPA's user avatar
  • 263
-1 votes
1 answer
168 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
-1 votes
1 answer
116 views

Continuous surjection of $\mathbb{R}^{n-1}$ onto the interior of the $n$-simplex with continuous right inverse

Let $n$ be a positive integer. Clearly $\mathbb{R}^{n-1}$ and the interior of the $n$-simplex $ \delta_n := \{x \in [0,1]^n:\,\Sigma_k x_n =1, (\forall i)\,x_i>0\} $ are homeomorphic. What I'm ...
ABIM's user avatar
  • 5,405
-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 ...
robert caro's user avatar
-1 votes
1 answer
148 views

Continuity of function mapping $\mathcal{P}(\mathcal{P}(X))$ to $\mathcal{P}(X)$ [closed]

Given a topological space $Y$, let $\mathcal{P}(Y)$ be the set of all probability measures on $Y$, endowed with the weak* topology. Let $X$ be a topological space (for convenience, it might be Polish ...
user66910's user avatar
-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
-1 votes
1 answer
75 views

Finiteness of "novel variance" from a kernel on a compact space [closed]

Let $c(i,i')$ be a kernel function on a reasonable index space $I$. Choose a dense sequence of points $\{i_1, i_2, \cdots \} \subseteq I$, and define the one-point kernel functions $k_n := c(\cdot, ...
Tom LaGatta's user avatar
  • 8,512
-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

1
5 6 7 8
9