All Questions
1,222 questions
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Is $L_p(X, \mu, E)$ uniformly convex for $p \in (1, \infty)$ if $E$ is a uniformly convex Banach space?
Let $(X, \Sigma, \mu)$ be a $\sigma$-finite complete measure space, $(E, |\cdot|)$ a Banach space, and $p \in (1, \infty)$. Let $L_p := L_p(X, \mu, E)$ be the Bochner space of all $\mu$-integrable ...
- 2,670
3
votes
1
answer
181
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General form of bounded linear functionals on Banach spaces
It is a famous result due to Riesz that every bounded linear functional $f$ on a Hilbert space $\mathcal{H}$ is of the form $f(x)=\langle x,z \rangle$ for a unique $z\in H$.
On p.188 of Introductory ...
- 6,367
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0
answers
55
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The initial sigma-algebra on the dual of a Banach lattice
Let $E$ be an AL space (i.e. a Banach lattice whose norm is additive on the positive cone $E_+$) that satisfies Mazur's condition (every sequentially weak$^*$-continuous functional on $E'$ is weak$^*$-...
- 12.9k
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0
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65
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Let $E$ be Banach, $\mu_n\to\mu$ weakly on a locally compact $X$, and $f \in C_b(X, E)$. Does $\int f\mathrm d\mu_n\to\int f\mathrm d\mu$ in norm?
Let
$X$ be a metric space,
$(E, |\cdot|)$ a Banach space
$\mathcal M(X)$ the space of all finite signed Borel measures on $X$,
$\mathcal C_b(X)$ be the space of real-valued bounded continuous ...
- 825
3
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3
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319
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Do these properties characterize Hilbert spaces?
Suppose $X$ is a Banach space with the following property: For any $x\in X$ there exists a two dimensional subspace $E$ isometric with $l_2^2$ such that $x\in E$. Does this property characterize a (...
- 4,786
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1
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68
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Is the Lorentz space $L_{W,1}(0,1)$ isomorphic to $L_1(0,1)$?
Let $W$ be a positive non-increasing continuous function on $(0,1]$ so
that $\lim_{t \rightarrow 0} W(t)=\infty$, $W(1)=1$ and $\int_0^1 W(t) dt =1$.
For $1 \leq p <\infty$, the Lorentz function ...
- 61
1
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1
answer
155
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Spectrum invariant under (generalised) transpose as operator on trace class operators
For matrices $A$ it is well known that the spectrum is invariant under transpose $\sigma(A^T) = \sigma(A)$. Furthermore, the spectrum of the adjoint matrix $\sigma(A^*) = \overline{ \sigma(A)}$ the ...
- 1,054
4
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1
answer
1k
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Projection exists ⇒ Uniformly convex?
I know that: Let X be a uniformly convex Banach space, $a\in X$ and $C\subset X$ closed and convex, then there is a unique $b\in C$ with $\left\Vert a-b\right\Vert=\inf_{x\in C}\left\Vert a-x \right\...
- 12.9k
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1
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244
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Can a weighted $\ell^p$ norm be bounded by an unweighted $\ell^q$ norm?
For any sequence $\omega\in[1, \infty)^{\mathbb{N}}$, define the weighted $\ell^p_\omega$-norm of the sequence $v$ by
$$\Vert v\Vert_{\ell^p_\omega} := \left(\sum_{k=1}^\infty \omega_k^p |v|_k^p\right)...
15
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5
answers
2k
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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 ...
- 63.9k
4
votes
0
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73
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The embedding of a Banach lattice in an ultrapower
Given a Banach space $X$ and a non-trivial ultrafilter $\mathcal{U}$ on a set $I$, the ultrapower $X_\mathcal{U}$ is defined as the quotient of $\ell_\infty(I,X)$ by the closed subspace $N_\mathcal{U}(...
- 4,461
81
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3
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Norms of commutators
If an $n$ by $n$ complex matrix $A$ has trace zero, then it is a commutator, which means that there are $n$ by $n$ matrices $B$ and $C$ so that $A= BC-CB$. What is the order of the best constant $\...
- 1,446
3
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0
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282
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Left ideals of $\ell^{\infty}(A)$ containing all weakly null sequences in a Banach algebra $A$
Let $A$ be a Banach algebra. $\ell^{\infty}(A)$, the space of all bounded sequences in $A$, is a Banach algebra with pointwise operations.
Let $w_0(A)$ be the subspace of all weakly null sequences in $...
CommunityBot
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2
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1
answer
184
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Example of a compact operator that is not uniformly continuous
I want to find a Banach space $E$ and a compact operator $K:[0,1]\times E \rightarrow E$ (that is, $K$ maps every bounded sequence onto a sequence that converges up to a subsequence) satisfying the ...
- 39.1k
4
votes
1
answer
169
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The complement of $L_1(0,1)$ in $L_1(0,1)^{**}$
Let $\mu$ be a finite measure, like the Lebesgue measure in $(0,1)$. It is well-known that $L_1(\mu)$ and its second dual $L_1(\mu)^{**}$ are Banach lattices, $L_1(\mu)$ is a projection band in $L_1(\...
- 61
13
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1
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401
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Is there a reflexive Banach space whose ball is not the convex hull of its extreme points?
Let $X$ be a reflexive Banach space. Then the convex hull of the extreme points of the unit ball is weakly dense by the Krein-Milman theorem and Kakutani's theorem. My question is, if there is an ...
- 109k
19
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1
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3k
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Infinite convex combinations in a Banach space
Let's say that a subset $C$ of a Banach space $X$ is $\sigma$-convex if the following property holds:
For any sequence $(x_k)_{k\ge0}$ in $C$, and for
any sequence of non-negative real numbers $(\...
- 2,821
5
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1
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364
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Isomorphic embedding of $l^n_{\infty}$ into $l_1^m$?
Given $n$, is there a $C(n)$-isomorphic embedding of $l^n_{\infty}$ into $l^m_1$ for sufficiently large $m$ and $C(n)<<\log(n)$?
For $n=2$ this can done with $m=2$. There are some results about $...
- 745
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0
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74
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Multiple steps of the Gorelik principle
The following result is one of several non-linear Banach space theory results known as The Gorelik Principle. I am stating it here in a weaker form than what is in the literature, but the statement ...
- 81
1
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1
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197
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About the constructions of Tsirelson space
Let $T'$ be the Tsirelson space dual in the Figiel-Johson construction, and $B_{T'}$ the unit closed ball. Then, $B_{T'}$ satisfies the next properties:
(i) $B_{T'}\subset B_{l^\infty}.$ Each vector ...
- 203
5
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1
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386
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Contact points for John's ellipsoid
Suppose $K$ is a centrally symmetric convex body in $\mathbb{R}^n$ and $E$ is the John's ellipsoid, the ellipsoid of maximal volume inside $K$.
If $E$ and $K$ have exactly $2n$ contact points, say $(\...
CommunityBot
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4
votes
1
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133
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A $C^*$ algebraic analogy of the concept of complemented subspace in the particular case of $\ell^\infty$
Let $A$ be a $C^*$ algebra. A $C^*$ subalgebra $C\subset A$ is said to be $C^*$ algebraic complemented of $A$ if there exist a $C^*$ subalgebra $D\subset A$ with $A=C\oplus D$ and the obvios mapping $...
- 42.8k
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1
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278
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A reproducing kernel Hilbert space
A functional Hilbert space $\mathscr H=\mathscr H(\Omega)$ is a Hilbert space of complex valued functions on a (nonempty) set $\Omega$, which has the property that point evaluations are continuous i.e....
- 25.8k
8
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1
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446
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Parallelogram law for vectors of equal length
Does the parallelogram law for vectors of equal length imply the full parallelogram law? That is,
if for all norm one vectors $x$ and $y$ in a Banach space $X$ it holds that $\lVert x-y\rVert^2+\lVert ...
- 1,453
4
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2
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339
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A space isometric to $\ell_\infty^2$
Consider a norm on $\mathbb C^2$ as $\|(z_1,z_2)\|:=\max\{|z_1|,|z_2|,\frac{1}{\sqrt{2}}|z_1+iz_2|\}.$
Question. Is $(\mathbb C^2,\|.\|)$ linearly isometric to $(\mathbb C^2,\|.\|_{\infty})$ where $\|(...
- 24.2k
0
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1
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269
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Determine if an integral expression is in $L^2(\mathbb{R})$
Note: This is a simplified version of the following question. I did not get a full response and realized can make it simpler to have my main interrogation answered. I decided to write it as a ...
8
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1
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1k
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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 $\...
- 42.8k
12
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1
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1k
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Uniform boundedness of an $L^2[0,1]$-ONB in $C[0,1]$
Assume that we have an orthonormal basis of smooth functions in $L^2[0,1]$. Are there useful practical criteria to determine whether the sup-norm of the basis functions has a uniform bound? I am sure ...
- 2,821
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0
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303
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Is Baire's theorem stronger than needed for functional analysis?
Many classic theorems in functional analysis involve using Baire's theorem to prove facts about topology that relate to maps between Banach spaces (or, more generally, F-spaces). The application ...
- 109
2
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0
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76
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Fractional integration in Orlicz spaces
I am reading the paper "Fractional integration in Orlicz spaces" by R. Sharpley.
And I would like to understand one question:
Let $A,B, C$ are Young's functions. The spaces $L_A, L_B$ are ...
- 63.9k
1
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1
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214
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An explicit description for a certain type of infinite-dimensional homogeneous polynomials
This is a side question from Infinite-dimensional "algebraic varieties".
Denote by $X_p$ ($1 \le p \le \infty$) the Banach spaces of complex sequences with finite $p$-norm and limit $0$. ...
- 1,543
1
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0
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61
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When is the metric on a Fréchet space homogeneous
Let $(F,d)$ be a Fréchet space over $\mathbb{F}\in \{\mathbb{C},\mathbb{R}\}$. Are there conditions under which, there exists some $C,d>0$ such that: for every $f\in F$ and every $k\in \mathbb{F}$ ...
- 6,367
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0
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140
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Extracting block sequences from $\ell_p^n$-like sequences in isomorphic copies of $\ell_p$
One of the characterizing properties of the canonical bases of the $\ell_p/c_0$ spaces is their perfect homogeneity (Zippin), so that every normalized block sequence in these spaces behaves like the ...
- 81
1
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0
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99
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Module homomorphisms modulo compact operators
Let $A$ be a Banach algebra. Let $L_a,R_a:A\to A$ denote the left/right multiplication operators $$L_ax = ax, \hspace{5mm} R_ax = xa$$ for all $a,x\in A$. Assume that no nonzero $L_a$ and $R_a$ is a ...
- 2,605
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164
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Quotients of $c_0$ that are complemented in $c_0$
Suppose $X$ is a closed subspace of $c_0$ with an unconditional basis and suppose also that it is a quotient of $c_0$. Is $X$ also a complemented subspace of $c_0$?
An affirmative answer implies that $...
- 1,073
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0
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81
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Example of the bounded convolution operator when Sharpley's conditions does not hold
I am reading about Orlicz and Marcinkevich spaces, and wondering whether there is an example in which Sharpley's condition is not satisfied for a special bounded operator $T_k$ (see for reference ...
- 97
1
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0
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97
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Are Hölder functions between Banach spaces residual in the compact-open topology?
Let $X$ and $Y$ be Banach spaces and let $C(X,Y)$ be the set of continuous functions from $X$ to $Y$ equipped with the topology of uniform convergence on compact sets (i.e. the compact-open topology). ...
- 5,405
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0
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39
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Example when Lorentz-Shimogaki condition satisfied with a specific Young's function
Let $\Psi(t)=\int_0^t\psi(s)$ be a Young's function. Then, $\Psi$ satisfies the Lorentz-Shimogaki condition if
$$
\int_0^{\infty}\frac{\Psi(st)}{v(t)^2}\psi(t)dt< \infty.
$$
Denote $\rho_{\Psi}=\...
- 6,367
3
votes
0
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257
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Complex Hölder space
I already posted this question on math.stackexchange, but got no response and was suggested to post it here.
I came across a space in an ergodic theory paper, which I am calling here a (complex) ...
- 136
4
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1
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244
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Boundedly complete bases
Let us recall that a basis $(x_{n})_{n}$ for a Banach space $X$ is boundedly complete if for every scalar sequence $(a_{n})_{n}$ with $\sup\limits_{n}\|\sum\limits_{i=1}^{n}a_{i}x_{i}\|<\infty$, ...
- 61.9k
4
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1
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405
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Why is it difficult to define a direct integral of Banach spaces or Banach algebras?
In the relevant Wikipedia entry, I can read about how to define a direct integral on Hilbert spaces and Von-Neumann algebras.
Suppose that I want to define a direct integral on either Banach spaces or ...
- 1,848
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0
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36
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Regarding significance of spectral variation under algebraic operations
I have been reading the paper Determining elements in $C^∗$-algebras through spectral properties.
The paper discusses about what would be the relation be between two elements $a$ and $b$ of a Banach ...
- 27
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5
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5k
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A criterion for the sum of two closed sets to be closed ?
Let $V$ and $I$ be two closed subsets of a Banach space $A$.
The set $V$ is a convex cone, and $I$ is a linear subspace of $A$. I also know that $V\cap I=\{0\}$.
I would like to know whether $I+V$ ...
- 225
1
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0
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85
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Interpolation between projective and injective spaces
Suppose $(\Omega,\mu)$ be a $\sigma$-finite measure space. Suppose $X$ is a Banach space and $L_p(\Omega;X)$ be the corresponding Bochner space for $0<p\leq\infty.$ Is it true that the complex ...
- 39.1k
1
vote
3
answers
580
views
Squeezing more convergence from the convergence in all $L^p$ spaces
Let $X$ be a space endowed with a finite measure $m$. Let $f_n : \to \mathbb C$ be measurable functions such that $|f_n| \le 1$ for all $n$ and $f_n \to 0$ in every space $L^p (X)$ with $p \in [1, \...
- 12.6k
7
votes
1
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200
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Projective tensor product of injective operators
I've seen claims that it is known that for a pair of bounded injective linear operators $T\colon X\to Y, S\colon W\to V$, their tensor product $T\otimes S\colon X \otimes_\pi W\to Y \otimes_\pi V$ ...
- 18.7k
1
vote
0
answers
111
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Properties of Sobolev spaces $W^{k,p}(\Omega, E)$ where $E$ is a Banach space
$\newcommand{\R}{\mathbb R}$Let $E$ be a Banach space with norm $\|\cdot\|_E$ and let $\Omega\subset \R^n$ be an open set.
For $k\geq 0, p\geq 1$ we define $W^{k,p}(\Omega, E)$, the Sobolev space of ...
- 2,533
4
votes
1
answer
490
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ODE in Banach space
Have I understood this correctly:
So originally we consider the following partial differential equation:
$$u_t= \frac{u_{xx}}{1+w}-\frac{1}{\epsilon}(1+w)u^3+\frac{wu}{\epsilon(1+w)} \text{ in } \...
- 41
5
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0
answers
169
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Is the Grassmannian of a Banach space an infinite dimensional manifold?
Grassmannian of complemented subspaces in a Banach space is a Banach manifold. This is explained for example in the thesis of Douady and is rather analogous to the finite-dimensional case.
I would ...
- 344
4
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1
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281
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Does property (V) imply the Grothendieck property for dual Banach spaces?
A Banach space $X$ has property (V) whenever for each Banach space $Y$, every unconditionally converging operator $T:X\to Y$ is weakly compact; equivalently, every non-weakly compact operator $T:X\to ...
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