All Questions
Tagged with gn.general-topology banach-spaces
91
questions
3
votes
1
answer
142
views
Approximating continuous functions from $K\times L$ into $[0,1]$
Let $K$ and $L$ be compact Hausdorff spaces, let $f:K\times L\to [0,1]$ be continuous and let $\varepsilon>0$. Can we find continuous $g_{1},...,g_{n}:K\to[0,+\infty)$ and $h_{1},...,h_{n}:L\to[0,+\...
- 5,275
13
votes
2
answers
660
views
Smooth Urysohn's lemma on Fréchet spaces
Let $V$ be a Fréchet topological vector space.
Let $K_0$ and $K_1$ be two closed subsets which are disjoint.
I wish to show the existence of a Fréchet-smooth function $f:V\to [0,1]$
whose restriction ...
- 42.2k
3
votes
1
answer
279
views
Pointwise convergence and disjoint sequences in $C(K)$
Let $K$ be a Hausdorff compact space and let $C(K)$ be the space of continuous real-valued functions on $K$. A sequence $(h_n)$ in $C(K)$ is called almost disjoint if there is a sequence $(g_n)$ with ...
- 5,275
0
votes
0
answers
41
views
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, \...
- 595
1
vote
0
answers
75
views
Morphism in commutative square strict?
Let $G,H$ be topological groups and $f:G\rightarrow H$ a continuous group homomorphism.
Then $f$ is said to be strict if $G/\mathrm{Ker}(f) \cong \mathrm{Im}(f)$ is an isomorphism of topological ...
- 463
7
votes
1
answer
753
views
Compactness of the unit ball of a Banach space for topologies finer than the weak* topology
Let $(\mathcal{X} , \|\cdot \|_\mathcal{X})$ be a Banach space and $\mathcal{X}'$ its topological dual. We denote by $\| \cdot \|_{\mathcal{X}'}$ the dual norm and define also the topological dual $\...
- 2,152
1
vote
0
answers
83
views
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). ...
- 5,001
15
votes
2
answers
903
views
Distinguishing topologically weak topologies of Banach spaces
Are the weak topologies of $\ell_1$ and $L_1$ homeomorphic?
Strangely may it sound, the question seeks contrasts between norm and weak topologies of Banach spaces from the non-linear point of view. ...
- 11.1k
1
vote
1
answer
213
views
An approximation property in a separable topological vector space
Let $X$ be a topological vector space.
Let us say that $X$ enjoys sequential separablity if there exists a sequence $\{x_n\}$ in $X$ such that for every $x\in X$ there exists a subsequence of $\{...
- 3,898
10
votes
2
answers
537
views
Implicit function theorem with continuous dependence on parameter
Let $X,Y$ be Hilbert spaces and $P$ a topological space$^1$ and $p_0\in P$.
Let $f:X\times P\to Y$ be a continuous map such that
for any parameter $p\in P$, $f_p:= f|_{X\times \{p\}}:X\to Y$ is ...
- 2,503
3
votes
0
answers
140
views
How can one construct this dendrite?
In the early 1970s Pelczynski noticed that the only surjective isometries on $C(K)$ for the following compact Hausdorff space $K$ are $\pm Id$. I believe this was the first such example.
Quoting from ...
- 1,063
5
votes
1
answer
240
views
How complex is the orbit equivalence relation of $\mathrm{Iso}_0(X)\curvearrowright S_X$ for $X=L^p([0,1])$?
For a Banach space $X$ let $S_X$ denote its unit sphere and let $\mathrm{Iso}_0(X)$ denote the group of rotations of $X$, that is isometries fixing the origin. There is a natural continuous action $\...
- 2,672
2
votes
1
answer
231
views
Is the union of good equivalence relations on a compact space good?
Let $X$, $Y_1$ and $Y_2$ be a compact Hausdorff spaces and let $\varphi_i:X\to Y_i$ be a continuous surjection (and so a quotient map).
Let $\sim$ be the minimal closed equivalence relation on $X$ ...
- 5,275
0
votes
1
answer
636
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\...
- 3
1
vote
1
answer
217
views
Is a topology sandwiched between two norms compactly generated?
Recall that a Hausdorf topological space $X$ is called compactly generated if any set whose intersections with compacts are compact is closed. Locally compact and first countable spaces are compactly ...
- 5,275
5
votes
1
answer
196
views
If a subspace $F$ is contained in a subspace $G$, and $H$ is close to $G$, can we choose a subspace of $H$ close to $F$?
Let $E$ be a Banach space. Recall that the collection of all closed linear subspaces of $E$ can be turned into a metric space in a number of ways. In particular, consider the notion of a gap: if $G$ ...
- 5,275
2
votes
0
answers
185
views
What is the smallest number of nowhere dense affine subsets covering a topological group?
$\DeclareMathOperator\cov{cov}\newcommand\A{\text A}$A subset $A$ of a group $G$ is called affine if $A=xHy$ for some subgroup $H\subseteq G$ and some $x,y\in G$.
Given a non-discrete topological ...
- 40.2k
1
vote
0
answers
1k
views
Weak sequential continuity vs strong continuity
Let $E$ be a Banach space, $T:E\rightarrow E$ a non-linear operator.
$T$ is said to be Weakly Sequentially Continuous (shortly W.S.C) on $E,$ if for every $\left(x_{n}\right)_{n}\subseteq E$ with $x_{...
- 291
1
vote
0
answers
694
views
A weakly sequentially continuous operator which is not weakly continuous
I'm reading some papers where the condition of weak sequential continuity is crucial instead of the weak continuity.
So, let
$T$ an operator between a Banach space $X$ and itself.
$T$ is weakly ...
- 291
2
votes
1
answer
177
views
Are there minimal topological conditions on a space 𝑋 for it to have a countable separating set?
Are there minimal topological conditions on a space $X$ for it to have a countable separating set?
A separating set here is a set $D \subset C(X)$ (where $C(X)$ is the space of continuous functions ...
- 339
2
votes
1
answer
67
views
Equicontinuity-like property of a convex compact set
Let $X$ be a Tychonoff topological space and let $x\in X$. Let $B\subset C(X)$ be convex and compact in the topology of pointwise convergence, and such that $f(x)=1$, for every $f\in B$.
Is there an ...
- 5,275
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 ...
- 5,001
0
votes
1
answer
75
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
$$
C_{\omega_i}(\mathbb{R}^n,\...
- 5,001
1
vote
0
answers
61
views
Connected components of bounded linear operators of $V = (\mathcal C(U(1), \mathbb C) , \lVert \cdot \rVert_\infty)$
This question is related to this one.
Consider the complex Banach space $V=(\mathcal C(U(1), \mathbb C), \Vert \cdot \Vert_\infty)$ where $\mathcal C(U(1), \mathbb C)$ is the space of continuous ...
- 1,355
0
votes
1
answer
78
views
If $\tau_1\subset \tau_2$ and $X^*$ is separable for $\tau_1$ then $X^*$ is separable for $\tau_2$?
Let $X$ be a Banach space the associated dual space is denoted by $X^*$. Take $\tau_1$ and $\tau_2$ two topologies in $X^*$ compatible with the duality $(X^*,X)$, such that $\tau_1\subset \tau_2$.
...
- 199
2
votes
0
answers
136
views
Is a closed connected semilattice of $C(I)$ path-connected?
Let $\Gamma $ be a sub-lattice of the Banach space $\big( B(S),\|\cdot\|_\infty\big)$ of all bounded real valued functions on the set $S$ (meaning that for any $f,g\in\Gamma $ both functions $f\wedge ...
- 55.5k
0
votes
1
answer
102
views
Breaking up dense subset in non-separable space
Let $X$ be a not necessarily separable (infinite-dimensional) Banach space and $D\subseteq X$ be dense linearly independent subset. Then does there exist a set of infinite-dimensional separable ...
- 5,001
3
votes
0
answers
143
views
Which metric spaces embed isometrically in $\ell_p$?
It is known that each metric space $X$ embeds isometrically in the Banach space
$\ell_\infty(X)$ of bounded (not necessarily continuous) functions $X \to \mathbb R$. Since $\ell_\infty(X)$ does not ...
- 1,761
0
votes
1
answer
113
views
$ \overline{(A-A)}\cap\overline{B}(0,r)\text{ is weakly compact, }\forall r>0 $?
Let $X$ be a separable Banach space and $A$ is a subset of $X$ such that
$$
A\cap\overline{B}(0,r) \text{ is weakly compact, } \forall r>0.
$$
Can we say that :
$$
\overline{(A-A)}\cap\overline{...
- 117
2
votes
2
answers
148
views
Can we say that : $ (A-B)\cap\overline{B}(0,r)\text{ is weakly compact, }\forall r>0 $
Let $X$ be a separable Banach space and $A,B$ are closed convex subsets of $X$ such that $B\subset A$ and
$$
A\cap\overline{B}(0,r) \text{ and } B\cap\overline{B}(0,r) \text{ are weakly compact, } \...
- 117
1
vote
1
answer
194
views
Density and the projective tensor product
Let $X$ be a locally convex space (over $\mathbb{R}$), $D\subset X$ be dense, $B$ be a Banach space (again over $\mathbb{R}$) with Schauder basis $\{b_i\}_{i =1}^{\infty}$. Is the set
$$
D^+\...
- 5,001
1
vote
1
answer
89
views
Metrization of quotient spaces defined by sequences of continuous functions
Let $K$ be a compact Hausdorff space and let $C(K)$ be the space of all scalar-valued continuous functions on $K$. Let $(f_{n})_{n}$ be a sequence in $C(K)$ satisfying $\sup\limits_{n}\sup\limits_{t\...
- 3,333
3
votes
1
answer
397
views
Criterion for weak convergence of sequences
Let $E$ be a normed space and let $F\subset E^{*}$. It is known that $F$ is dense if and only if the restriction of $\sigma(E,F)$ on $B_E$ coincides with the weak topology.
Hence, if $F$ is dense and ...
- 5,275
0
votes
0
answers
141
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 ${...
- 245
1
vote
0
answers
27
views
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 $f(x)\ne 0$...
- 5,275
4
votes
1
answer
85
views
Approximation of multipliers by multipliers of a smaller set
Let $X$ be a compact metric space, and let $B$ be a convex balanced bounded set in $C(X)$ such that for every $x\in X$ there is $f\in B$ with $f(x)\ne 0$.
Let $M=\{u\in C(X),~ uf\in B,~\forall f\in B\...
- 5,275
5
votes
1
answer
606
views
Can $L^1_{loc}$ be represented as colimit?
Let $L^1_{loc}$ denote the set of all functions from $\mathbb{R}$ to itself which are locally integrable. For every infinite compact subset $K\subseteq \mathbb{R}$, let $L^1_{m_K}$ denote the space ...
- 5,001
4
votes
1
answer
169
views
A map into a Hilbert space with prescribed orthogonality
Let $X$ be a locally compact separable metric space, and let $L:X\times X\to \mathbb{C}$ be continuous and such that $L(x,x)=1$ and $L(y,x)=\overline{L(x,y)}$, for every $x,y$.
Does there always ...
- 5,275
2
votes
1
answer
116
views
Size of the orbit of a dense set
This question is a follow-up to: this post.
Let $X$ be a separable Banach space, $\phi\in C(X;X)$ be an injective continuous non-affine map, and $A$ be a dense $G_{\delta}$ subset of $X$. How big ...
- 63
0
votes
2
answers
318
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
5
votes
0
answers
221
views
What is the smallest number of hyperplanes covering $\ell_2$?
For a Banach space $X\ne \{0\}$, let $\mathrm{cov}_H(X)$ be the smallest number of hyperplanes covering $X$.
By a hyperplane in a Banach space I understand any closed affine subspace of codimension ...
- 40.2k
2
votes
1
answer
206
views
Relation between the weak star topology and hereditary Lindelöfness
Let $X$ be a Banach space. Is the following implication valid?
$$ (X,w) \textrm{ is hereditarily Lindelöf}~ \Rightarrow X^*~ \textrm{is separable} $$
The converse is clearly true, since the ...
- 3,898
4
votes
0
answers
113
views
point-wise approximation of the identity in hereditary Lindelof spaces
Let $X$ be a topological vector space. Assume that there exists a sequence of finite range measurable functions $\phi_n:X\to X$ with $\lim\phi_n(x)=x$.
Q. Can we concluded that $X$ is hereditery ...
- 3,898
10
votes
1
answer
355
views
Are all compact subsets of Banach spaces small in a measure-theoretic sense?
Definition. A subset $K$ of a topological group $X$ is called measure-continuous if there exists a $\sigma$-additive Borel probability measure $\mu$ on $X$ such that for every compact subset $C\subset ...
- 40.2k
9
votes
1
answer
364
views
Comparing two $\sigma$-algebras on $B(\ell^1)$
Let us consider $B(\ell^1)$, bounded linear operators on $\ell^1$. We recall the weak operator topology, denoted by $w$, on $B(\ell^1)$ is determined as follow
$$w-\lim T_i=T \Longleftrightarrow \...
- 3,898
4
votes
1
answer
365
views
Separable Lindelöf locally convex spaces that are not second-countable
A Lindelöf space is a topological space in which every open cover has a countable subcover.
Does there exists a Lindelöf locally convex space which is not second countable?
I am also looking for a ...
- 3,898
6
votes
1
answer
412
views
Is each compact metric space a subset of a compact absolute 1-Lipschitz retract?
A metric space $X$ is called an absolute $L$-Lipschitz retract if for any metric space $Y$ containing $X$ there exists a Lipschitz retraction $r:Y\to X$ with Lipschitz constant $Lip(r)\le L$.
...
- 40.2k
8
votes
1
answer
324
views
Why $S$ cannot be homeomorphic to the $1$-sphere of $\ell^2$?
Consider the $\ell^2$ complex Hilbert space.
Let $m\in \mathbb{N}^*$ be a fixed number, and set
$$
S=\left\{ x=(x_n)_n\subset \ell^2\ :\ \sum_{n=1}^m \frac{|x_n|^2}{n^2}=1\right\}.$$
I want to ...
- 724
8
votes
1
answer
507
views
Are the following subsets of a Hilbert space always homeomorphic?
Let $F$ be a infinite-dimensional complex Hilbert space, with inner product $\langle\cdot\;| \;\cdot\rangle$, the norm $\|\cdot\|$, the 1-sphere $S(0,1)=\{x\in F;\;\|x\|=1\}$ and let $\mathcal{B}(F)$ ...
- 724
5
votes
2
answers
789
views
Covering compactness in the weak sequential topology
Let $X$ be a real Banach space. Apart from the norm topology, we can consider the following weak topologies on $X$:
the weak toplogy, defined as the initial topology with respect to $X^*$. In other ...
- 171