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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 ...
C_Al's user avatar
  • 251
15 votes
1 answer
601 views

Topological spaces in which countable intersections of dense open sets have dense interior

In certain topological spaces, known as Baire spaces (e.g., completely metrizable spaces), a countable intersection of dense open sets is dense. Now consider the following strengthening of the Baire ...
Gro-Tsen's user avatar
  • 32.5k
1 vote
0 answers
76 views

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,\...
Gaspar's user avatar
  • 161
6 votes
1 answer
500 views

A characterization of metric spaces, isometric to subspaces of Euclidean spaces

I am looking for the reference to the following (surely known) characterization of metric spaces that embed into $\mathbb R^n$: Theorem. Let $n$ be positive integer number. A metric space $X$ is ...
Taras Banakh's user avatar
  • 41.9k
7 votes
1 answer
331 views

A metric characterization of Hilbert spaces

In the Wikipedia paper on Hadamard spaces, it is written that every flat Hadamard space is isometric to a closed convex subset of a Hilbert space. Looking through references provided by this Wikipedia ...
Taras Banakh's user avatar
  • 41.9k
13 votes
2 answers
2k views

When can we divide continuous functions?

Let $X$ be a compact Hausdorff topological space such that for every continuous $f,g:X\to\mathbb{R}$ with $0\le f\le g$ there is a continuous $h:X\to\mathbb{R}$ such that $f=gh$. What can be said ...
erz's user avatar
  • 5,529
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
13 votes
2 answers
915 views

Topological vector spaces (reference request)

In his book Topological Function Spaces Arhangel'skii says that "it is well known that every nontrivial locally convex linear topological space $X$ is homeomorphic to a space of the form $Y \...
Peluso's user avatar
  • 674
3 votes
0 answers
80 views

Every Borel linearly independent set has Borel linear hull (reference?)

I am looking for a reference to the following fact, which probably is known and could be proved somewhere by someone. Theorem. The linear hull of any linearly independent Borel set in a Polish ...
Taras Banakh's user avatar
  • 41.9k
3 votes
1 answer
951 views

Specific criterion for the sum of two closed sets to be closed

Let $Y$ and $Z$ be two closed subspaces of a Banach space $X$ with $Y\cap Z=\{0\}$. I know that $Y+Z$ is a closed subspace of $X$ $\iff \exists \alpha > 0:\quad \lVert y\rVert \le \alpha\lVert y+z\...
Westlife's user avatar
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
2 votes
0 answers
134 views

Stone–Weierstrass theorem for stronger topologies

The Stone–Weierstrass theorem gives an easy to check criterion on a (algebra) set of functions $D\subseteq C(X)$ which ensures that $D$ is dense. Are any similar results for density in $C_b(X)$ ...
ABIM's user avatar
  • 5,405
1 vote
0 answers
104 views

"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 ...
ABIM's user avatar
  • 5,405
2 votes
1 answer
941 views

Metrizability of topology of compact convergence

Let $X$ be a separable metric space and $Y$ be a second-countable $\sigma$-compact Hausdorff space. Then the compact-convergence (compact-open) topology on $C(Y,X)$ is metrizable with metric $$ d(f,g)...
ABIM's user avatar
  • 5,405
2 votes
1 answer
348 views

Topological spaces containing paths

Let $C(\mathbb{R}^n;\mathbb{R}^d)$ be the space of continuous functions with the uniform-convergence on compacts topology. What are function spaces $X$ building on $C(\mathbb{R}^n;\mathbb{R}^d)$? $X$...
ABIM's user avatar
  • 5,405
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
7 votes
1 answer
1k views

Reference request: norm topology vs. probabilist's weak topology on measures

Let $(X,d)$ be a metric space and $\mathcal{M}(X)$ be the space of regular (e.g. Radon) measures on $X$. There are two standard topologies on $\mathcal{M}(X)$: The (probabilist's) weak topology and ...
JohnA's user avatar
  • 710
4 votes
0 answers
105 views

Borel selections of usco maps on metrizable compacta

The problem posed below is motivated by this problem of Chris Heunen and in fact is its reformulation in the language of usco maps. Let us recal that an usco map is an upper semicontinuous compact-...
Taras Banakh's user avatar
  • 41.9k
2 votes
1 answer
266 views

Approximate selection for finite-valued upper hemicontinuous/semicontinuous maps?

I'd like to know if there are any known-results on the existence of continuous approximation theorems for upper hemicontinuous (aka upper semicontinuous) maps $\phi: X\rightarrow Y$ which are finite ...
Fred's user avatar
  • 85
5 votes
0 answers
330 views

The second dual of $C(X)$ with the compact-open topology

Let $X$ be a compact Hausdorff space. Then $C(X)$ is a Banach algebra with the supremum norm and so is $A=C(X)^{**}$ under either Arens product. Moreover, it is easy to verify that $A\cong C(Z)$ for ...
user124775's user avatar
1 vote
2 answers
223 views

Is $C_b(Q,E)$ linearly isometrically isomorphic to $C(\beta Q,E)$ where $\beta Q$ is the Stone–Čech compactification of $Q$?

Let $Q$ be a locally compact Hausdorff space and $E$ be a Banach space. Let $C(Q)$ be the collection of all real-valued continuous functions on $Q$ and $C_b(Q,E)$ be the collection of all $E$-valued ...
Idonknow's user avatar
  • 623
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
1 answer
381 views

Sufficient criteria for $X \subset \mathcal{H}$ to be a Lipschitz (or unif. cont.) retract of $\mathcal{H}$

I am interested in sufficient criteria which ensure that a subset $X$ of a Hilbert space $\mathcal{H}$ is a Lipschitz (or at least uniformly continuous) retract of $\mathcal{H}$. Under which ...
PhoemueX's user avatar
  • 734
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 ...
Henry.L's user avatar
  • 8,071
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 ...
Henry.L's user avatar
  • 8,071
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 ...
Jairo Bochi's user avatar
  • 2,479
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 $X_1$. ...
JumpJump's user avatar
  • 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 ...
JumpJump's user avatar
  • 679
15 votes
5 answers
2k views

Between Tietze's and Dugundji's extension theorems

The celebrated Tietze extension theorem asserts that any continuous real-valued function defined on a closed subset of a normal space, can be extended to a continuous function on the whole space. Seen ...
Pietro Majer's user avatar
  • 60.5k
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
233 views

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 ...
Włodzimierz Holsztyński's user avatar
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
4 votes
0 answers
2k views

Approximation of continuous functions by Lipschitz functions in the topology of uniform convergence on compact sets

I was involved into this subject when I answered this question from MSE. Trying to generalize my answer, I am thinking about a following Question. Let $X$ and $Y$ be metric spaces. When each ...
Alex Ravsky's user avatar
  • 5,409
2 votes
1 answer
891 views

Riesz representation theorem for vector-valued fields

Let $Q$ be a locally compact Hausdorff space, and let $V$ be a topological vector space. Consider the space $X = C_0(Q, V)$ of $V$-valued fields which vanish at infinity. Let $X^*$ denote the dual ...
Tom LaGatta's user avatar
  • 8,512
3 votes
1 answer
502 views

Determining continuous functions on Banach spaces

Let $X$ be a real Banach space. For a continuous (not necessarily linear) function $g:X \to \mathbb{R}$ and a family $\mathcal{F} \subseteq X^*$, we´ll say that $\mathcal{F}$ determines $g$ if ...
Ramiro de la Vega's user avatar
8 votes
1 answer
1k views

Ring of continuous functions, reference request.

I am looking for a reference for the following facts in functional analysis and topology. (If these "facts" are not true, I suppose I'm looking for the closest approximation which is true.) Let $X$ ...
Marty's user avatar
  • 13.3k
9 votes
1 answer
1k views

A question concerning separate and joint continuity of bilinear maps

Suppose that $V$ is a locally convex topological vector space and $f:V^2 \to V$ is a bilinear map. Suppose that $C \subseteq V$ is compact and convex, $f$ maps $C^2$ into $C$ and $f \restriction C^2$ ...
Justin Moore's user avatar
  • 3,547
3 votes
0 answers
251 views

What is the origin of the metrization problem for compact convex sets?

The following is an ``old question in analysis:'' Is it true that every perfectly normal compact convex subset of a locally convex topological vector space is metrizable? Here perfectly normal means ...
Justin Moore's user avatar
  • 3,547
13 votes
1 answer
736 views

Idempotent measures on the free binary system?

Let $(S,*)$ be the free (non associative) binary system on one generator (so $S$ is just the set of terms in $*$ and $1$). There is an extension of $*$ to the space $P(S)$ of finitely additive ...
Justin Moore's user avatar
  • 3,547