Questions tagged [banach-spaces]

A Banach space is a complete normed vector space: A vector space equipped with a norm such that every Cauchy sequence converges.

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Strong differentiability and the inverse function theorem in Banach spaces

I am trying to prove the strong differentiability version of the Inverse Function Theorem for Banach spaces, but I am not sure if it is true. I am interested in this because it is a kind of punctual ...
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127 views

Banach spaces whose biduals are $L_{1}$

Let $X$ be a Banach space. If $X^{**}$ is linearly isometric to $L_{1}(\mu)$ for some $\sigma$-finite measue $\mu$, we shall say that $X$ is an $L_{1}$-pre-bidual. Question 1. What are the examples of ...
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242 views

Characterization of differentiability

For a normed space $(V, \lVert\cdot\rVert_V)$ let us define: \begin{equation} \forall x, y \in V \quad [0,1] \mapsto \gamma_x^y (t) = (1-t)x + ty. \end{equation} I would like to ask whether the ...
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Banach spaces whose second conjugates are separable

It was known that the James space $J$ has separable second conjugate, is non-reflexive and isometric to its second conjugate. I want to know whether there are Banach spaces $X$ with separable second ...
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50 views

On weakly precompact sets

Recall that a bounded subset $A$ of a Banach space $X$ is said to be weakly precompact if every sequence in $A$ admits a weakly Cauchy subsequence. Rosenthal's $l_{1}$-theorem states that a bounded ...
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119 views

A characterization of the Dunford-Pettis property

A Banach space $X$ is said to have the Dunford-Pettis property if for any Banach space $Y$ every weakly compact operator $T:X\rightarrow Y$ is completely continuous. Recall that $T$ is completely ...
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40 views

Leray-Schauder degree in Banach manifolds

The so called Leray-Schauder degree is usually defined for maps of the form $I - f$, where $f: X \to X$ is a compact map defined on a Banach space. Is there an extended definition for the setting of ...
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109 views

Convex pigeonhole principle in Banach spaces

In this question, all Banach spaces will be infinite-dimensional and separable, and all subspaces will be infinite-dimensional and closed. Say that a subset of the unit sphere $S_X$ of a Banach space $...
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100 views

Second dual $X^{**}$ of ternary $C^*$-ring $X$ is again ternary $C^*$-ring?

Recall that a ternary $C^*$-ring is a complex Banach space $X$, equipped with a associative ternary product $[.,.,.]:X^3 \to X$ which is linear in outer variables and conjugate linear in middle ...
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108 views

Hereditarily primary Banach spaces

A Banach space $X$ is said to be prime if every infinite dimensional complemented subspace is isomorphic to the space $X$. The space $X$ is primary if it has an infinite dimensional subspace $Y$ such ...
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128 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 ...
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103 views

Which group is the standard group of isometries?

For the classical sequence spaces $\ell_p$ ($p\not=2$) and $c_0$ each surjective linear isometry $U$ has the form $U(a_i)=(\varepsilon_i a_{\pi(i)})$ for a permutation $\pi$ of $\mathbb{N}$ and $\...
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207 views

Surjective linear isometries on $\ell_\infty(\mathbb{N})$

In Volume 1 of "Classical Banach Spaces" Lindenstrauss and Tzafriri note that all surjective linear isometries on $\ell_\infty$ are of the from $(a_i) \mapsto (\varepsilon_i a_{\pi(i)})$ ...
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1answer
162 views

Trying to understand construction of $C^*$-algebra corresponding to a ternary $C^*$-ring from a paper

Recall that a ternary $C^*$-ring is a complex Banach space $X$, equipped with a associative ternary product $[.,.,.]:X^3 \to X$ which is linear in outer variables and conjugate linear in middle ...
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1answer
104 views

Showing non-injectivity [closed]

Let $X,Y$ be Banach spaces and let $A:X\rightarrow Y$ be a linear operator. Does it suffice to show that there exists a sequence $x_n\in X$ such that $\lim_{n\rightarrow\infty}Ax_n = 0$ with $||x_n||=...
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103 views

Must a Schauder basis for $W^{1,p}_0(\Omega)$ be oscillatory?

Suppose that $\Omega \subset \Bbb R^d$ is a sufficiently nice domain. From the examples of orthogonal bases in Hilbert space cases (or looking at a wavelets basis), it seems natural to me that one may ...
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1answer
70 views

weak*-null sequences in the dual space of a separable space

Let $X$ be a separable space and let $x^{**}\in X^{**}$. If $x^{**}(x^{*}_{n})\rightarrow 0$ for each weak*-null sequence $(x^{*}_{n})_{n}$ in $X^{*}$, is $x^{**}$ in $X$ ? Thank you!
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On weak Hahn-Banach smoothness

Let us recall Phelp's property-$U$: A subspace $Y\subset X$ is said to have property-$U$ if every $y^*\in Y^*$ has unique norm preserving extension over $X$. $Y$ is weak Hahn-Banach smooth if $y^*$ ...
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Regarding uniform continuity of a function on a C*- algebra

Let $A$ be a complex unital C*- Algebra. Let $f:A\longrightarrow\mathbb{C}$ be a function(not necessarily linear) which is continuous and $|f(a)|\leq\|a\|$ for every $a\in A$. Then can we say that $f$ ...
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160 views

UMD constant of finite dimensional spaces

For a Banach space $B$, its one-sided Unconditional Martingale Difference (UMD) constant $C^-_p$ (for $p \in (1,\infty)$) is the smallest value such that for all $B$-valued martingale difference ...
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155 views

Self-dual Orlicz sequence spaces

Suppose that $\ell_\phi$ is a reflexive Orlicz sequence space such that its dual space $\ell_\phi^*$ is isomorphic to $\ell_\phi$. Is $\ell_\phi$ isomorphic to $\ell_2$?
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249 views

Connectedness of Invertible elements in a non- commutative C*- algebra

The Gelfand Naimark Segal theorem says that any complex C* algebra $A$ is isometrically isomorphic to a C* sub-algebra of bounded operators on a Hilbert space. Here we see that the set of all ...
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93 views

Construction of Schauder bases on $C(X)$

Let $(X,d)$ be a compact metric space and let $C(X)$ be the set of continuous (bounded) real-valued functions on $X$ equipped with the usual supremum norm: $$ \|f\|_{\infty}\triangleq \sup_{x\in X}|f(...
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99 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^...
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Are stable images closed?

If $X$ is a Banach space and $T : X \to X$ is a continuous linear operator with the property that $T^{n}X$ equals $T^{n+1}X$ for some $n \ge 1$, does it follow that $T^{n}X$ is a closed subspace?
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136 views

Operator in the commutant which is small on a given vector

Suppose $x$ is a non-zero vector in a Banach space, and $T$ is a fixed operator. Is the following true: For any $\varepsilon, \delta$, there exists $S$ in the commutant of $T$ such that $1\leq\|S\|<...
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46 views

Gateaux derivative of operator semigroup

Let $X$ be a Banach space and $(T(t))$ be an analytic semigroup generated by $A$. Let $B$ be an operator such that $F(\varepsilon) = A + \varepsilon B$ generates a semigroup $S^{\varepsilon}_t$ for ...
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61 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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1answer
131 views

Closure of the space of Fredholm operators

Let $X,Y$ be two Banach spaces. A bounded operator $A$ is Fredholm if $\ker A$ and $\mathrm{coker} A$ are finite dimensional. Denote by $Fred(X,Y) \subset \mathcal{L}(X,Y)$ the space of Fredholm ...
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174 views

Differentiability of the map $x\mapsto \delta_x$ in the Arens-Eells/Lipschitz-free space

$\DeclareMathOperator\AE{AE}\DeclareMathOperator\Lip{Lip}$Let $\AE(X)$ denote the Arens-Eells space on a Banach space $X$. Consider the map: $$ \begin{aligned} \delta: X & \rightarrow \AE(X) \\ x&...
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77 views

Functions with smooth projections on finite-dimensional subspaces

Let $E,F$ be Banach spaces and $F$ be finite-dimensional and $E$ be strictly convex. Let $f\in C(F,E)$ have the property that: $$ \text{For every finite-dimensional subspace $E'\subseteq E$ we have } ...
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131 views

Standard function spaces with the approximation property

A Banach space $\mathcal{X}$ is said to have the approximation property (AP) if, for every compact set $K \subset \mathcal{X}$, there is a sequence of finite rank operators $\{U_n : \mathcal{X} \to \...
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1answer
217 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 $\...
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1answer
238 views

Banach space with dual not a GT space

Let $X$ be a Banach space. A bounded linear map $u:X\to\ell_2$ is said to be $1$-summing if for all finite sequence $(x_i)\subseteq X$ there is a constant $C>0$ such that $\sum\|ux_i\|\leq C\sup\...
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64 views

Need reference of books/papers which deals with Ternary Banach Algebras

I'm interested in learning about ternary Banach Algebras ( mainly ideal theory and tensor product) Can someone please recommend me some papers/ books/ notes which deals with mentioned topics? Thank ...
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1answer
76 views

Banach space containing uniformly complementend $\ell_p^n$s

Let $X$ be a Banach space such that both $X$ and $X^*$ have finite cotype. Also assume that $X$ is an inductive limit of finite dimensional Banach spaces $X_n\subseteq X_{n+1}.$ Fix $1<p<\infty.$...
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1answer
181 views

Duality $(M/N)^*\equiv N^\perp/M^\perp$ for closed subspaces $N\subset M$ of a Banach space

Let $M$ be a closed subspace of a Banach space $X$. Then we can identify $(X/M)^*$ with $M^\perp$ and $M^*$ with $X^*/M^\perp$. Indeed, if $Q^*:X\to X/M$ is the quotient map, then $Q^*:M^*\to X^*$ is ...
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1answer
99 views

Operators "building" linear independant sets

Let $E$ be a separable Banach space and let $T\in L(E,E)$. Is there a condition on $T$ ensuring that: $$ \mbox{$\{x_n\}_{n=1}^N\subseteq E$ is linearly independent} \Rightarrow \{T(x_n)\}_{n=1}^N\cup \...
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121 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 $L_\infty(0,1)$?...
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148 views

Subprojective Orlicz sequence spaces

A Banach space $X$ is subprojective if every infinite dimensional closed subspace $Y$ of $X$ contains an infinite dimensional subspace $Z$ which is complemented in $X$. I am interested in conditions ...
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101 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 ...
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1answer
71 views

Semi-norms on LCS inductive limit of Banach Spaces

Let $(E_n,i_n)_{n\in\mathbb{N}}$ be an direct system of Banach spaces in the category of locally convex spaces (LCSs) with continuous linear maps and let $E_{\infty}$ by their inductive limit. What ...
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1answer
143 views

Reference request: Baire's theorem for operator ranges

Let $F$ be a Banach space. A vector subspace $U \subseteq F$ is called an operator range if there exists a Banach space $E$ and a bounded linear mapping $T: E \to F$ such that $TE=U$. By a quotient ...
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72 views

Hamel basis with all coordinate functionals discontinuous

If $X$ is an infinite dimensional (separable) Banach space, can one find a Hamel basis $(x_\alpha)_{\alpha\in\Lambda}$ such that all coordinate functionals $x_\alpha^*(x_\beta)=\delta_{\alpha,\beta}$ ...
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1answer
61 views

Regarding an element being self adjoint

Let $A$ be a unital C*-algebra. Let $x,y\in A$ be self adjoint elements in $A$, with $x$ being invertible. Can we say that the spectrum of $x^{-1}y$ is a subset of the real line? I know this is true ...
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1answer
147 views

Biorthogonal weakly null basic sequences

Let $E$ be a Banach space, let $e_{n}\in E$ and $g_{n}\in E^{*}$ be biorthogonal basic sequences (i.e. $\left<e_n,g_m\right>=\delta_{mn}$ ). Moreover, both of these sequences are weakly null. (...
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1answer
151 views

Some estimates on tensor norms

Denote $M_n$ to be $n\times n$ matrix. For $X\in M_n$ define $\|X\|_1:=\max\limits_{1\leq j\leq n}\sum_{i=1}^n|x_{ij}|$ and $\|B\|_2:=\max\{|\sum_{i,j=1}^nb_{ij}x_iy_j|:|x_i|=|y_j|=1,\ 1\leq i,j\leq n\...
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1answer
102 views

Uniform boundedness principle for almost surely converging sequence of operators

I'd like to do the following: I consider a separable Banach space $X$ with a probability measure $\mu$ on the Borel $\sigma$-algebra $\mathcal B(X)$. Additionally, I have a sequence of measurable, ...
3
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1answer
68 views

G.L. l. u. st. for subspaces of Banach spaces with an unconditional basis

A Banach space $X$ has Gordon-Lewis local unconditional structure (G.L. l. u. st.) if for every finite dimensional subspace $E$ of $X$, the inclusion operator $i:E\to X$ factors through a finite ...
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1answer
148 views

$K$-convex Banach spaces

Let $X$ be a Banach space. We say that $X$ contains $\ell_1^n$'s uniformly iff for all $n\in\mathbb N$ there exist subspaces $X_n\subseteq X$ with $d(X_n,\ell_1^n)\leq \lambda$ for some $\lambda\geq 1$...

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