All Questions
Tagged with banach-spaces fa.functional-analysis
1,222 questions
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47
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Is the embedding $i: (L^p_\text{loc} (Y), \| \cdot \|_{L^p_\text{loc}}) \to (L^0(Y), \hat \rho)$ continuous or Borel measurable?
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, \...
- 657
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0
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113
views
The set of measurable functions together with convergence in measure is a completely metrizable abelian topological group
Below we use Bochner measurability and Bochner integral. Let
$(X, \mathcal A, \mu)$ be a complete $\sigma$-finite measure space,
$(E, | \cdot |)$ a Banach space,
$S (X)$ the space of $\mu$-simple ...
- 657
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0
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77
views
Property (H) in the dual norm
Consider the Hilbert space $l_2$ with an equivalent norm
$$\Vert x \Vert = \max \{2 \Vert x \Vert_1, \Vert x \Vert_2 \},$$
where $\Vert x \Vert_1 =( \sum_{n=2}^\infty x_n^2 )^{\frac{1}{2}}$ and $\Vert ...
- 85
0
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1
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145
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Renorming on a separable Banach space
Let us consider the sequence space $c_0$ with the equivalent norm
$$\Vert x \Vert^2 = \max_{i\ge1} \vert x^i \vert^2 + \sum_{i=2}^{\infty} 2^{-i+1} \vert x^i \vert^2 $$
for $x=(x^1,x^2,\ldots)\in c_0$....
- 85
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0
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208
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Examples of non $w^{*}$-closed complemented subspaces of a dual Banach space that are also dual spaces
Let $Y$ be a complemented, but not $w^{*}$-closed, subspace of a Banach space $X$. It is known that certain such $Y$ are not dual spaces.
Question: What are interesting examples of subspaces of the ...
- 7,067
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1
answer
92
views
Is $\Lambda:= \pi_2 \circ \pi_1:E \to L$ surjective?
Let $E$ be a Banach space. For a linear map $T$, we denote by $R(T)$ its range and by $N(T)$ its kernel. Let $I:E \to E$ be the identity map. Let $T:E \to E$ be a compact (bounded linear) operator. ...
- 657
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114
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Norm distance in a Banach space
Consider the Hilbert space $l_2(\mathbb{N})$ under the square summable norm $\Vert \cdot \Vert_2.$ Let us define a new norm $||| \cdot ||| $ equivalent to $\Vert \cdot \Vert_2$ such that the closed ...
- 85
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141
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Does there exists a smooth reflexive infinite dimensional Banach space that is not strictly convex
It is known that in a reflexive Banach space, if the norm is strictly convex, then its dual will be smooth Banach space, and if the norm is smooth, then the dual norm is strictly convex.
We can find ...
- 85
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127
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Closure of BV paths in space of paths of finite $p$-variation
Let $p\ge1$ and $T>0$. Define $\mathscr D([0,T])$to be the space of partitions of $[0,T]$, where each partition is a finite collection of distinct points of $[0,T]$. Consider a continuous path $X:[...
- 177
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168
views
Completely continuous maps from projective tensor products into $c_0$
Let $E$, $F$ be two Banach spaces and $E\mathbin{\hat{\otimes}}_{\pi}F$ denote their projective tensor product.
For each unit norm $\xi\in E$ and $\gamma\in F$, let's define
$$
J_{\gamma}:E\to E\...
- 2,605
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0
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92
views
Finding set of best approximations from a point in $c_0$ to its subspace
Given $X$=$c_0$, null sequence space with sup norm. Consider a subspace $Y$ of $c_0$ consisting of elements of $c_0$ as, $Y=\{x\in c_0 : x_{2i}=i.x_{2i-1}, i \geq 1\}$. I need to find the set of best ...
- 85
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1
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232
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A generalization about the density of $\mathcal C_c(X, E)$ in $\mathcal L_p (X, \mu, E)$ when $E$ is a Banach space
Let $X$ be a metric space, $\mu$ a $\sigma$-finite non-negative Borel measure on $X$, and $(E, |\cdot|)$ a Banach space. Let $\mathcal L_p := \mathcal L_p (X, \mu, E)$ and $\|\cdot\|_{\mathcal L_p}$ ...
- 657
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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 ...
- 835
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303
views
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
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36
views
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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241
views
Can a non-reflexive space embed into a reflexive space?
My question is inspired from the concept of super-reflexivity which was defined by James here: https://www.cambridge.org/core/journals/canadian-journal-of-mathematics/article/superreflexive-banach-...
- 11
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103
views
A question on the Haar basis for $L_{1}[0,1]$
Let $(x_{n})_{n=1}^\infty$ be a basis for a Banach space $X$. It is important to know the exact expression of the norm of $\|\sum_{i=1}^{n}a_{i}x_{i}\|$ for all $n$ and all scalars $a_{1},a_{2},\ldots,...
- 3,343
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56
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Existence of minimal subset of dual ball such that the intersection of kernels is trivial
Let $(V, \lVert \cdot \rVert)$ be a separable Banach space and let $B_{V^*}$ denote the closed ball in the dual $V^*$. Suppose we have a family $C \subseteq B_{V^*}$ such that $\bigcap_{\Lambda \in C} ...
- 820
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109
views
Operator algebra on an invariant subset
In Rickart, page 50 Theorem 2.2.1, the statement is made: A linear subspace $\mathfrak{M}$ of the algebra $\mathfrak{A}-\mathfrak{L}$ is invariant with respect to the representation $a{\rightarrow}A_a^...
- 109
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291
views
Operator norm on tensor product of trace classes is multiplicative
Given Hilbert spaces $\mathcal H_1,\mathcal H_2,\mathcal K_1,\mathcal K_2$ and bounded linear maps $S_i:\mathcal B^1(\mathcal H_i)\to\mathcal B^1(\mathcal K_i)$, $i=1,2$ between the respective trace ...
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171
views
Is $\ell_2$ is isomorphic to a subspace of $L_\infty(0,1)$?
I know that $\ell_2$ is isomorphic to a subspace of $L_p(0,1)$ for any $1\le p<\infty$. However, I haven't seen anything about $L_\infty$. Is $\ell_2$ is isomorphic to a subspace of $L_\infty(0,1)$?...
- 617
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0
answers
161
views
When does a positive operator preserve invertibility
Let $\Omega_1,\Omega_2$ be compact Hausdorff spaces and let $P:C(\Omega_1)\longrightarrow C(\Omega_2)$ be a unital positive operator. I wanted to know if there is a necessary and sufficient condition ...
- 175
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answers
67
views
Dual of isometric copies into dual Banach spaces
Let $X$ be a Banach space and $X_1\xrightarrow{}X$ isometrically. Under some assumption can we guarantee that $X^*$ contains an isometric copy of $X_1^*$. I am also interested to know if this happens ...
- 1,879
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129
views
Certain decompositions of decomposable Banach spaces
Let $\mathcal{X}$ be a decomposable Banach space (i.e. a topological direct sum of infinite-dimensional subspaces, say $\mathcal{X}=\mathcal{A}\oplus\mathcal{B}$). Can one always obtain another ...
- 1,453
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168
views
Sequence of functions tending to zero in L^2
Let us consider a sequence of functions $f_n : (0,1)\times (0,1) \to \mathbb{R}$ in $L^2((0,1)\times (0,1))$ satisfying the following condition:
$$
\lim_{n \rightarrow \infty}\int_{1/j}^{1 - 1/j}\...
- 29
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0
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302
views
Convergence of characteristic functions vs. weak convergence of measures and the Ito-Nisio theorem
In section 2.6 of Linde's Probability in Banach Spaces: Stable and Infinitely Divisible Distributions the author is pointing out that in infinite-dimensional Banach spaces the convergence of ...
- 167
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0
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106
views
A noncontinous algebra map between Banach algebras
What is an example of two Banach algebras $A$ and $B$, and an algebra map $\phi:A \to B$ which is not continuous?
- 203
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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 ...
- 5,405
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0
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68
views
Closed graph theorem for cones?
In the paper "A strong open mapping theorem for surjections from cones onto Banach spaces, Marcel de Jeu and Miek Masserschmidt, Adv. Math." it is proved (among other things) that if $X, Y$ are ...
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0
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154
views
Use of this space of very rapidly decreasing continuous functions
Let $C_n$ denote the subspace of continuous function on $[0,\infty)$ supported on $[n,n+1]$. Denote the $\ell^p$-direct sum Banach space
$$
V_p
:=
\left\{
f \in C([0,\infty)):\,
\sum_{n=1}^{\infty} ...
- 5,405
0
votes
1
answer
126
views
Tauberian operators
Let $X$ be a Banach space non reflexive and $T$ from $l_2(X)$ to $l_2(X)$ a bounded operator defined by:
$$T(x_n )=\frac{x_n }{n}.$$
We know that :
$$T^{**-1}(l_2(X))=\{x_n^{**} \in l_2(X^{**}) : \...
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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 ${...
- 307
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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
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0
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58
views
Bounds on $\inf_{x,x' \in \mathbb B_X}TV(P+x,Q+x')$, where $P$ and $Q$ are distributions with density on the space $X=(\mathbb R^n,\ell_p)$
Let $n \ge 1$ be an integer, $p \in [1,\infty]$, and $P$ and $Q$ be two (probability) measures on the metric space space $X=(\mathbb R^n,\ell_p)$ which have densities w.r.t the Lebesgue measure on $X$,...
- 6,853
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0
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113
views
Extrapolate an Interpolation scale
Suppose $X$ and $Y$ are real Banach spaces with a continuous embedding $X\subset Y$. For given $0<\theta<1$ I am interested in constructing using the norms of $X$ and $Y$ a (Quasi-) Banach ...
- 23
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0
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97
views
Does $L^p$ contractivity imply $L^p$ dissipativity?
Does $L^p$ contractivity of an operator semigroup imply the $L^p$ dissipativity of the operator ?
Thank you in advance !
- 1
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0
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59
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Nests on Banach spaces and their duals
Let $X$ be a Banach space and $\mathcal{E}$ a nest on $X$.
Take $f\in X^{*}$ and suppose:
$N \in\mathcal{E}$ is the largest element of the nest so that $f \in N^\bot$
$N=\bigcap_{M>N}M$
Is there ...
0
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0
answers
115
views
If two spheres are isometric, does there exist a bijective isometry $T:S\to S$ with $\|Tu-\alpha Tv\|_Y \leq \|u-\alpha v\|_X$ for all $\alpha>0?$
Let
$$(S,\|\cdot\|) = \{(x,y)\in \mathbb{R}^2: \|(x,y)\| =1\},$$
that is, $S$ is the collection of all norm one vectors in $\mathbb{R}^2$ with respect to the norm $\|\cdot\|.$
Question: Let $\|\...
- 623
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161
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Topologies corresponding to norm, SOT and WOT under duality
This is a question from MSE which has not received any attention so far.
Let $X$ be a Banach space with norm dual $X'$. (I am mostly interested in the case $X = \ell^1$.)
For a linear mapping $T : X \...
- 1,773
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0
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977
views
Weak convergence can imply strong convergence [duplicate]
In $\ell^1(\mathbb N)$, weak convergence implies strong convergence. Is there a classification of infinite-dimensional Banach spaces for which such a property holds true ?
- 16.2k
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0
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70
views
A question about an irreducible ultra-power
Let $A$ be a Banach algebra and $E$ be an irreducible Banach $A$-module. Is there a countably incomplete ultra filter $\mathcal U$ on $\mathbb N$, the set of natural numbers, such that the ultra power ...
- 2,118
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votes
0
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55
views
Continuity of a composite function
Let $n=2$ or 3 and let $\Omega$ be a bounded domain of $\mathbb{R}^n$. Let $T>0$ and $f \in L^2([0,T],H^1(\mathbb{R}^n))$.
Is the mapping
\begin{equation}
\begin{array}{rcl}
C^0([0,T],C^1(\bar{\...
- 143
0
votes
1
answer
328
views
Find the trace for some elements in group algebra
Let $K=\langle b,c,d\mid b^{2}=c^{2}=d^{2}=bcd=1\rangle $. Now we consider $$D=K*\mathbb Z/2\mathbb Z=\left\{a,b,c,d\mid a^{2}=b^{2}=c^{2}=d^{2}=bcd=1\right\}$$ where $*$ is the free product. Then we ...
- 407
0
votes
0
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65
views
Does $\{ x^* \circ \psi_t:x^*\in ext(E^*), t\in K \}\subset ext(X^*)$ hold?
Notations: Let $K$ be a locally compact Hausdorff space and $E$ be a real normed linear space.
Recall that $C_0(K,E)$ is the set of $E$-valued continuous functions $f$ on $K$ such that $f$ vanishes at ...
- 623
0
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0
answers
57
views
A question on order unbounded sequences in Banach lattices
Let $E$ be a Banach lattice. It is well-known that every norm convergent sequence in $E$ admits an order convergent subsequence and hence admits an order bounded subsequence. But it seems that a norm ...
- 3,343
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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 ...
- 5,529
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votes
0
answers
263
views
Does AX+XA=0 have any non-trivial solutions?
Let $X$ be a continuous linear self-adjoint operator on some Hilbert space $H$ and for arbitrary compact operators $A$ we have: $XA+AX=0.$ Does this imply that $X=0$ or can there be non-trivial ...
- 305
0
votes
0
answers
59
views
Restriction to Basis of Cadlag function
If $f \in L^2([0,T])$ then it can be written as
$$
f(t) \triangleq \sum_{i \in \mathbb{N}} c_i e_i(t),
$$
for some sequence $\{c_i\}$ of real numbers and a Schauder basis $\{e_i(t)\}$ of $L^2([0,T])$ ...
- 5,405
0
votes
0
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123
views
On the operators from $l_{p}$ into Tsirelson's space $T$
Let $1<p<2$. My question is: Is any operator from $l_{p}$ into Tsirelson's space $T$ compact?
- 3,343
0
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0
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115
views
When do block sequences yield disjoint subspaces?
Let $X$ be a Banach space having a (unconditional, normalized) Schauder basis $(e_n)_n$. Suppose that $Y$ and $Z$ are (closed) block subspaces of $X$ having normalized block bases (with respect to $(...
- 3,115