All Questions
1,222 questions
4
votes
1
answer
86
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\...
- 5,529
5
votes
1
answer
158
views
What is a name for co-Sobczyk Banach spaces?
Definition. Let us define a Banach space $X$ to be co-Sobczyk if every linear bounded operator $T:Z\to c_0$ defined on a separable subspace $Z$ of $X$ extends to a bounded operator $\bar T:X\to c_0$.
...
- 42k
2
votes
0
answers
129
views
Logical axioms used in the construction of counterexamples to ISP
In many cases, some problems are either solved in an affirmative way, or in a negative way. However, in some cases it turns out that some logical axioms lead to a proof of a certain statement, while ...
- 1,175
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 $\tau_1\subset \tau_2$.
...
- 199
5
votes
4
answers
4k
views
Non-separable Banach space
The vector space $C_b(\mathbb R)$ of bounded continuous functions on $\mathbb R$ is non-separable: it is possible to produce a direct proof of this fact, mimicking the standard proof for the non-...
- 16.2k
4
votes
1
answer
137
views
Are unit balls in Banach spaces retracts of bidual balls?
Let $X$ be a separable Banach space embedded canonically in $X^{**}$. Is there a retraction from the unit ball $B_{X^{**}}$ of $X^{**}$ onto the unit ball $B_X$ of $X$?
When we insist on uniformly ...
- 97
11
votes
3
answers
445
views
Does the generator of a 1-parameter group of Banach space isometries know which elements are entire?
Let $X$ be a complex Banach space. Let $(\sigma_t)_{t \in \mathbb{R}}$ be a 1-parameter group of linear isometries of $X$ which is strongly continuous i.e. $t \mapsto \sigma_t(x)$ is continuous for ...
- 662
2
votes
3
answers
256
views
Compact restrictions of the inclusion of $J:L_\infty(0,1)\to L_1(0,1)$
Given Banach spaces $X$, $Y$ and a bounded operator $T:X\to Y$ with non-closed range, a perturbation argument shows that there exists an infinite-dimensional closed subspace $M$ of $X$ such that the ...
- 4,461
4
votes
2
answers
244
views
Compact images of nowhere dense closed convex sets in a Hilbert space
Let $B=-B$ be a nowhere dense bounded closed convex set in the Hilbert space $\ell_2$ such that the linear hull of $B$ is dense in $\ell_2$.
Question. Is there a non-compact linear bounded operator ...
- 42k
1
vote
0
answers
64
views
Embedding a normed space as a hyperplane
Let $X$ be a real normed space and suppose that $X$ is a closed hyperplane of a bigger space $\tilde X$. Given any unit vector $u$ in $\tilde X\setminus X$, consider the function $p:X\to\mathbb R$ ...
- 483
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,\...
- 5,405
11
votes
1
answer
953
views
Separable bidual but nonseparable third dual
Does there exist a Banach space $X$ such that $X^{**}$ is separable but $X^{***}$ is non-separable?
More generally, for every natural $n$ can someone construct an example of Banach space $X$ such ...
- 521
-1
votes
1
answer
320
views
Existence of weak limit for bouded sequence $\{y_n\}$ such that for every weak limit point $\{y_n\}$ must equal $y$
Let $X$ be separable Banach space and $\{x_n\}$ be a bounded sequence, relatively weakly compact sequence in $X$. we set $y_n=\frac{1}{n}\sum_{i=1}^{n}{x_i}$, then (together with the Krein and ...
- 199
2
votes
1
answer
70
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 ...
- 5,529
7
votes
0
answers
124
views
The bidual of the space of divergence-free vector fields
Consider the Banach space $L_1(\mathbb R^n, \mathbb R^n)$ of integrable vector fields $(n>1$) together with its subspace $N$ formed by those vectors fields whose divergence (computed in the ...
- 11.3k
3
votes
3
answers
358
views
Preannihilators of subspaces of separable duals
If $Y\subset X^*$ is a closed subspace (where $X$ is a separable Banach space), the preannihilator of $Y$ in $X$ is $Y_{\perp}:=\{x\in X : y^*(x)=0, \forall y^*\in Y \}$. If $Y$ is a proper subspace ...
- 1,361
6
votes
2
answers
735
views
Tensor product space with projective norm is incomplete
Ryan says in his book "Introduction to Tensor Products of Banach Spaces"(pg. 17) that for Banach spaces $X$ and $Y$, $X\otimes Y$ equipped with projective norm is not complete unless $X$ and $Y$ are ...
- 163
8
votes
1
answer
268
views
Two questions about basic sequences
Suppose $(x_n)$ and $(y_n)$ are two basic sequences in a separable Banach space $X$ such that $\overline{span}\{(x_n), (y_n)\}=X$. Can we always pass to subsequences $(x_{n_k})$ and $(y_{n_k})$ such ...
- 355
17
votes
1
answer
912
views
$(1+\epsilon)$-injective Banach spaces, complex scalars
It is well known that a real Banach space which is $(1+\epsilon)$-injective for every $\epsilon >0$ is already 1-injective (Lindenstrauss Memoirs AMS, 1964, download here). Using common ...
- 1,704
1
vote
0
answers
139
views
Any reflexive space has the property of Banach-Saks?
We say that a Banach space $(X,\|.\|)$ has the Banach-Saks property if every bounded sequence $(x_m)_m$ in $X$ admits a subsequence $(x_{m_n})_n$ which converges in the sense of Cesàro, that is, there ...
- 115
4
votes
1
answer
191
views
Intrinsic volumes of non-polyconvex, non-compact sets
I am reposting this question I asked and bountied on Math SE, which has been upvoted but not answered or commented on.
The intrinsic volumes (AKA Minkowski Functionals or, with different ...
- 509
6
votes
3
answers
3k
views
Why isn't the theorem of approximation applicable in Banach spaces?
Let X be a Hilbert space, A a convex, closed subset of X. Then there exists for every x in X exactly one best approximation in A, that is there exists a y in A such that || x-y || = d(x,A) = inf { || ...
- 61
30
votes
3
answers
3k
views
Surjectivity of operators on $\ell^\infty$
Can anyone give me an example of an bounded and linear operator $T:\ell^\infty\to \ell^\infty$ (the space of bounded sequences with the usual sup-norm), such that T has dense range, but is not ...
- 301
3
votes
1
answer
273
views
Predual to Lipschitz maps with $p$ derivatives
Let $p\in \mathbb{N}$, and define $\mathrm{Lip}_p$ be the collection of functions from $\mathbb{R}^d$ to itself, with $p-1$ first derivatives bounded and whose $p^{\mathrm{th}}$ derivative is ...
- 5,405
10
votes
0
answers
230
views
Norm-attaining operators with values in a 2-dimensional Hilbert space
Is the set $N\!A(X,\ell_2^2)$ of norm-attaining operators from a Banach space $X$ onto the $2$-dimensional Hilbert space $\ell^2_2$ dense in the Banach space $L(X,\ell_2^2)$ of all linear continuous ...
- 4,535
13
votes
4
answers
2k
views
Is the category of Banach spaces with contractions an algebraic theory?
Consider the category of Banach spaces with contractions as morphisms (weak, so $\|T\| \le 1$). Is this an algebraic theory?
I suspect that this is true. The "operations" will be weighted sums, ...
- 26.8k
7
votes
1
answer
1k
views
weak*-closed subspaces
Recall that a closed subspace $Y$ of a Banach space $X$ is weakly complemented if the set
$$Y^{\bot}:= \{ f\in X^*| f(y) = 0 \forall y\in Y\}$$
is a complemented subspace of $ X^*$. For example, $c_0$ ...
- 195
4
votes
1
answer
388
views
Condition for Banach space that the distance function can always reach its minimum at a unique point
Let $(X,\|\cdot\|)$ be a Banach space and $B\subset X$ be a bounded subset. Now define a function $f_B:X\to [0,\infty)$ by $f_B(x)=\sup_{b\in B}\|b-x\|$.
My question is in which kind of Banach space,...
- 155
0
votes
2
answers
1k
views
Does point-wise weak convergence give weak convergence in $L^2(I;X)$?
Let $X$ be a separable reflexive Banach space, $F$ be a locally Lipschitz nonlinear operator on $X$ that is weakly continuous on $X$, and $u_n$ are $u_n$ weakly converges to $u$ on $L^2(0,T;X)$. Now, ...
- 395
0
votes
1
answer
178
views
Convergence in LB-spaces
Edit:
Let $X$ be a strict LB-space described by $\lim X_n$ and suppose that $\{x_n\}_{n \in \mathbb{N}}$ converges in $X$. I'm looking for a reference showing that $x_n$ must converge in some $X_N$.
- 5,405
10
votes
1
answer
366
views
Are all compact subsets of Banach spaces small in a measure-theoretic sense?
Definition. A subset $K$ of a topological group $X$ is called measure-continuous if there exists a $\sigma$-additive Borel probability measure $\mu$ on $X$ such that for every compact subset $C\subset ...
- 42k
8
votes
1
answer
325
views
Why $S$ cannot be homeomorphic to the $1$-sphere of $\ell^2$?
Consider the $\ell^2$ complex Hilbert space.
Let $m\in \mathbb{N}^*$ be a fixed number, and set
$$
S=\left\{ x=(x_n)_n\subset \ell^2\ :\ \sum_{n=1}^m \frac{|x_n|^2}{n^2}=1\right\}.$$
I want to ...
- 724
15
votes
3
answers
2k
views
Disintegrations are measurable measures - when are they continuous?
This is a sequel to another question I have asked.
The notion of disintegration is a refinement of conditional probability to spaces which have more structure than abstract probability spaces; ...
- 8,512
4
votes
0
answers
75
views
What are the complemented subspaces of $(\bigoplus\ell_q^n)_p$?
Bourgain/Casazza/Lindenstrauss/Tzafriri proved in their unconditional basis UTAP book (1985) that $\ell_1$ is the only nontrivial complemented subspace of $(\bigoplus\ell_2^n)_1$, and hence by duality ...
- 1,591
1
vote
1
answer
394
views
Are $\| \Delta u \|_{L^p(\Bbb R^d)} + \| u \|_{L^p(\Bbb R^d)}$ and $\| u \|_{W^{2,p}(\Bbb R^d)}$ equivalent norms?
Is it true that $$\| \Delta u \|_{L^p(\Bbb R^d)} + \| u \|_{L^p(\Bbb R^d)}\quad\text{and}\quad\| u \|_{W^{2,p}(\Bbb R^d)}$$ are equivalent norms?
This results is pretty easy and straightforward for $...
- 1,101
2
votes
0
answers
83
views
Integral convergence with two sequences of functions
I came across this theorem just stated but has not proved and marked by 'it is easy to see'.
Theorem If $u_m$ and $v_m$ converges to $u$ and $v$ in $L^2([0,T];H^1(\Omega))$ weakly and $L^2([0,T];L^2(\...
- 239
1
vote
0
answers
73
views
The $w^{*}$-convergent sequences and the Mackey topology on $X^{*}$
Let $X$ be a Banach space. Recall that the Mackey topology $\mu(X^{*},X)$ on $X^{*}$ is the topology of uniform convergence on weakly compact subsets of $X$. Let $(x^{*}_{n})_{n}$ be a sequence in $X^{...
- 3,343
8
votes
1
answer
523
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
1
vote
1
answer
303
views
Density of norm-attaining operators
By Bishop-Phelps theorem we know that for a real Banach space, the set of all norm attaining bounded linear functionals is norm-dense in $X^*$, the topological dual of $X$. We also know that in ...
- 585
-1
votes
1
answer
108
views
Are local diffeomorphisms Fredholm maps with index zero? [closed]
Does this statement correct? if it does how we can prove it. In Banach spaces
a map is local diffeomorphism if and only if it is a Fredholm map of index zero with no critical points?
- 71
13
votes
1
answer
724
views
Trace-class operator satisfies $\sum |\lambda_n|<\infty$?
Here's an "exercise" which I thought should be easy, but which I find myself unable to do.
Let $V$ be a Banach space.
Recall that an operator $f:V\to V$ is trace-class if it is in the image of the ...
- 43.2k
0
votes
1
answer
208
views
Can a hyperplane be contained in a subspace?
Suppose $Y$ is a subspace of a normed linear space $X$ and let $y\in S_Y, y^*\in S_{Y^*}$ such that $y^*(y)=1$, where $S_Y$ denotes the closed unit sphere in $Y$. My question is the following:
Is it ...
- 585
3
votes
3
answers
2k
views
Determining if a set is a Basis for l^2
For each $ n\ge 1$ Define the vectors $e_n = (e_{nk})$ where $ k\ge 1$ and $ e_{nk} = \frac{1}{k^n}$
Is this set a basis for $l^2$?
Thanks,
- 4,153
0
votes
1
answer
102
views
Law of a step function and its generalization to two dimensions on an appropriate spaces
Let's consider two discontinuous functions defined on $D$ and $D \times [0,T]$, respectively:
A step function: $u_1(x)=\begin{cases}
u_{L}, x<c_1, \\[2ex]
u_{R}, x>c_1,
\end{cases}$
A "...
- 657
2
votes
1
answer
378
views
Does the norm on a sequence space have to be monotone?
Let $\rho:[0,+\infty)^{\mathbb{N}}\to[0,+\infty] $ satisfy the following properties:
$\rho(\lambda u)=\lambda\rho( u)$, for every $u\in [0,+\infty)^{\mathbb{N}}$ and $\lambda\ge 0$;
$\rho(u+v)\le \...
- 5,529
0
votes
1
answer
131
views
A question on the metric approximation property
Let $X,Y$ be Banach spaces. Suppose that $X^{***}$ has the metric approximation property. Let $T:X^{**}\rightarrow Y$ be a finite-rank operator and let $\epsilon>0$. Is there a finite-rank operator ...
- 3,343
8
votes
0
answers
330
views
Complementability of finite dimensional subspaces
Suppose $X$ is a separable infinite dimensional Banach space, $E$ a finite dimensional subspace which is $c$-complemented in $X$. Is the following true?
For any $\varepsilon>0$, one can find $x\...
- 1,361
6
votes
2
answers
3k
views
Closed convex bounded sets are weakly compact for which spaces?
It is known that for all reflexive Banach spaces, closed convex bounded sets are weakly compact (compact for the weak topology).
What is the general class of topological vector spaces for which this ...
- 549
0
votes
0
answers
170
views
Limit of balls in $L^p$
Setup:
Let $\mu$ be a measure on a measurable space $(X,\Sigma)$, such that for every $p ,q\in [1,\infty)$, $L^p_{\mu}(\Sigma)\subseteq L^q_{\mu}(\Sigma)$ if $p\geq q$. Furthermore, the inclusions ...
- 5,405
6
votes
1
answer
450
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 $Lip(r)\le L$.
...
- 42k