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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If a series has the same sum under any rearrangement, then is it absolutely convergent?

Let $(V,\| \cdot \|)$ be a Banach space. Let $\{a_n\}$ be a sequence in $V$ such that $\sum a_n$ converges. Assume that for every bijection $f:\mathbb{N}\rightarrow \mathbb{N}, \sum a_n = \sum ...
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12 views

non-reflexive Banach space

I am having some difficulties with part of a problem, I am working on. Let $X$ be a non-reflexive Banach space and let $i: X \to X^{**}$ be the canonical embedding. Show that for given $\epsilon > ...
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39 views

Showing a set is nowhere dense in $C([0,1])$

Let $E_n$ be the set of all $f \in C([0,1])$ for which there exists $x_0 \in [0,1]$ (depending on $f$) such that \begin{align*} |f(x)-f(x_0)| \leq n|x-x_0| \end{align*} for all $x \in [0,1]$. Why is ...
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1answer
29 views

About composition of Holder functions.

Let $f,g$ be Holder continuous functions with respective exponents $\alpha, \beta \in (0,1)$. More precisely $f \in C^{\alpha}(\mathbb{R}^n;\mathbb{R}^n)$, $g\in C^{\beta}(\mathbb{R}^n,\mathbb{R})$. ...
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20 views

Get locally uniformly convex norm by bounded linear operator

I want to prove this theorem Let $Y$ be a locally uniformly convex Banach space and $X$ be a Banach space and let $T$ be a bunded linear operator from $X$ into $Y$ such that for every bounded ...
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37 views

Modulus of convexity

My question has already been asked here: Link I am sorry for not seeing this before.
2
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1answer
26 views

Kernel of the Extension of a Bounded Linear Operator

Suppose $T\colon E\to F$ is a bounded linear operator between Banach spaces. Moreover let $i\colon E\to E’$ be a dense, compact inclusion of $E$ into some other Banach space $E’$. Finally assume that ...
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1answer
47 views

Borel Measure on Banach Space

While thinking about what some measure on an infinite dimensional Banach space could look like a came across the point that if I'd like to assign a size to all epsilon balls, they by Riesz' lemma ...
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1answer
43 views

Book suggestion geometry of Banach spaces

I am studding geometry of Banach spaces and applications in metric fixed point theory. Does anyone have a good recommendation of books//lectures/resources/etc.? Thanks.
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34 views

Is Banach space a correct context to study sequences and series?

Recently, I have reviewed elementary analysis and I realized that every theorem in the text(Rudin-PMA) about series can be generalized to Banach space. Here is an example. Below is the theorem ...
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44 views

How to show that $e^{tA}=\frac{1}{2\pi i}\int_{\{Re \ \lambda =a\}}e^{\lambda t}(\lambda I-A)^{-1}d\lambda$?

Let $X$ be a Banach space and $A:X\to X$ be a bounded operator. We can show that if $|\lambda|>|A|$ then $\lambda I-A$ is invertible and $$(\lambda ...
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1answer
32 views

On a Banach space $X$, is the functional $x \mapsto \frac{1}{p}\|x\|^p$ convex?

Let $X$ be a Banach space. Let $p > 1$ and, consider the functional $X \to \mathbb{C}$ given by: $$x \mapsto \frac{1}{p}\|x\|^p$$ I would like the know if the above functional is convex. That ...
3
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27 views

Every Cauchy net is convergent [duplicate]

Prove that in a Banach space every Cauchy net is convergent. I have trouble to prove this, please help.Thanks Edit:Let $A$ be a directed set and $\{f_{\alpha}\}_{\alpha\in A}$ is a net in $X$ ...
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19 views

Baillon theorem in fixed point theory

Baillon proved that if $X$ is a Banach space with $\rho'_X(0)<\frac{1}{2}$, then $X$ has the fixed point property (by $\rho_X(t)$ we denote the modulus of smoothness). My questions are as ...
2
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2answers
39 views

Banach Space: Open Unit Ball Totally Bounded?

Just to be sure: In an infinite dimensional Banach space the open unit ball cannot be totally bounded, right? The context is that I need this in order to find a lack in here...
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1answer
106 views

Radon-Riesz & Kadec-Klee

Let us say that a normed vector space has the a) RR (Radon-Riesz) property if for any sequence, norm convergence is equivalent to weak convergence + convergence of norms. b) KK (Kadec-Klee) property ...
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1answer
45 views

Show that C is a closed convex subset and its element of minimum norm

I have a lot of problems with the following exercise that I can't solve. Let $(L^1((0,1)), \|\cdot\|_{L^1}):=(E,\|\cdot\|)$ and $$C:=\{u\in E:u\geq0 \text{ a.e } x\in (0,1),\quad T(u)\geq 1\},$$ ...
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1answer
45 views

Spectrum of a finite rank operator

If $ T\in B(H)$ is a finite rank operator, then there are orthonormal vectors $e_1,...,e_n$ and vectors $g_1,...,g_n$ such that $Th=\sum_{i=1}^n (h,e_i )g_i$, then we can easily see that $T$ is ...
4
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1answer
80 views

Proof that $(L^1)\neq(L^\infty)^\ast$

I have seen a "proof" that $L^1\neq(L^\infty)^\ast$ which goes as follows: show that there is an element of $(L^\infty)^\ast$ which is not in the image of the canonical map ...
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1answer
28 views

Let $A: X \to X$ be a Fredholm operator, then $Ax=y$ has a solution iff $Ax=0$ implies $x=0$?

Let $X$ be a Banach space and let $A: X \to X$ be a Fredholm operator, then $Ax=y$ has a solution iff ($Ax=0$ implies $x=0$)? I can't see how this is implied by the common definitions of Fredholm ...
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33 views

Do the radii of a family of nested balls (in a Banach space) converge?

I apologize for the stupid question, but I am getting a bit crazy about this. Consider a Banach space $X$ and a sequence of nested closed balls $(B_n)_n$, i.e. $B_{n+1} \subset B_n$. Let $r_n$ ...
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1answer
35 views

Show that $C^1([0,1])$ is not reflexive

Aim of this exercise is proving that $(C^1([0,1]),\|\cdot\|_{C^1})$ is not reflexive. We know that, if $(f_h)_h\subset C^1([0,1])$ is a sequence that weakly converges to $f\in C^1([0,1])$ (that is ...
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1answer
32 views

What type of Banach spaces $X$ does the sum $x + c$ make sense where $x \in X$ and $c \in \mathbb{R}$?

What are such spaces called where we can add a constant to an element of the Banach space and the addition makes sense somehow? Eg. in $L^2$ this always is sensible. Is there a difference to the name ...
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26 views

Topological isomorphism vs isometric isomorphism

We say that: $T:(X,\|\cdot\|_X)\rightarrow (Y,\|\cdot\|_Y)$ is a isometric isomorphism if it is a linear isomorphism, and it is an isometry, that is $\|T(x)\|_Y=\|x\|_X\quad \forall x\in X;$ ...
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27 views

give necessary and sufficient conditions that every functional in $w^*-cl M$ be the $w^*$- limit of a sequence from M

Let $X$ be a separable Banach space. If $M$ is a linear manifold in $X^*$ give necessary and sufficient conditions that every functional in $w^*-\mathrm{cl} M$ be the $w^*$- limit of a sequence from ...
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1answer
14 views

Does “uniformly isolated” imply closed?

Let $X$ denote a complete metric space and consider a subset $A \subseteq X$. Call $A$ uniformly isolated iff there exists $r > 0$ such that for all $a \in A$, we have that $B_r(a) \cap A = \{a\}$. ...
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25 views

difference between uniformly convex norms and strictly subadditive norms?

What is the difference between uniformly convex norms and strictly subadditive norms? why we need to define two above concept? how they help us to study Banach spaces? Is the norm induced by ...
3
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1answer
21 views

convex weak* sequentially closed subset of a separable Banach space implies weak* closed

I'm studying Conway's a course in Functional Analysis by myself. The following is corollary 6.12.7 of this book. If $X$ is a separable Banach space and $A$ is a convex subset of $X^*$ that is weak* ...
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33 views

Adjoint of an operator on $C(X)$

Let $X,Y$ be compact and $\tau: Y\to X$ be continuous. If $A:C(X)\to C(Y)$ such that $(Af)(y)=f(\tau(y))$ is a linear and continuous operator, then show that the adjoint operator $A^*:M(Y)\to M(X)$ is ...
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1answer
51 views

Maximal subspace on which an operator is bounded

Consider the Banach space $X=C[0,1]$ of real continuous function on $[0,1]$ equipped with the supremum norm. Consider the operator $A:D(A)\to X$, $Af=f'$ for each $f\in D(A)=C^1[0,1]$. We can see that ...
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59 views

Properties of reflexive Banach spaces

I just want to see the importance of reflexive Banach spaces and what is special about them compared to other Banach spaces. What kind of properties hold in reflexive spaces that do not necessarily ...
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basic sequence in the complexification induces a basic sequence in the underlying real space?

This should be easy to prove if it is true, but, alas, what SHOULD be easy is not always easy for me ;) Conjecture 1. Let $X$ be a real Banach space and let $X_\mathbb{C}$ denote its ...
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66 views

Show that a subspace of l2 is not complete

I would like to know if this exercise is correct. Let $\Bbb R^\infty=\{x:\Bbb N\rightarrow \Bbb R: \exists n \text{ such that}\quad x(k)=0 \quad \forall k\geq n\}$. Show that $(\Bbb R^\infty, \| ...
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32 views

Show that an operator is bounded.

Let $\{\alpha_{mn} ;m,n\geq 1\}$ be scalars satisfying a- $M=\sup_n\sum_{m\geq 1}|\alpha_{mn}|<\infty $ , and b- $\sup_n|\alpha_{mn}|<\infty$, then $(Af)(n) = \sum_{m\geq 1}\alpha_{mn} f(m)$ ...
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61 views

When is $M+N$ closed

Let $X$ be a Banach space and $M,N$ be closed subspaces. If the range of linear transformation $x\to (x+M)\oplus (x+N)$ from $X$ into $X/M\oplus X/N$ is closed show that $M+N$ is closed. or using ...
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1answer
26 views

Power Series for a Banach Contractive Mapping function

For each continuous function $f \in C[0,1]$ define the continuous function $T(f)(x)$ by $$T(f)(x)=x^2 + \int_{0}^{x}tf(t)dt$$ for each $x\in C[0,1]$. I'm trying to find the power series to represent ...
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Reinventing The Wheel - Part 2: The Lebesgue Integral

Disclaimer After struggling for some time to find an appropriate definition for the notion of integration I came across another attempt for which I would need your help deciding to what extend this ...
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42 views

Strongly continuous semigroup of operators which cannot be extended to a group

Let $X$ be a Banach space. We call a family of bounded operators $(T(t))_{t\in \mathbb{R}}$ a strongly continuous group if it satisfies the properties of the strongly continuous semigroup but for ...
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22 views

weak* continuous linear functional

I studied Megginson's An introduction to Banach space theory. I saw below proposition without any proof; Proposition:Let X be a normed space. Then a linear functional on X* is weakly* continuous if ...
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23 views

Helffer-Sjöstrand-Formula: Idea behind?

I have to present the Helffer-Sjöstrand-Formula. Now I'm wondering: Why does it include a factor $\chi(y\langle x\rangle^{-1})$ for some bump function $\chi$ and the chinese symbol ...
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36 views

Restriction of an operator on a Banach space

Let $X$ be a Banach space and $A:D(A)\to X$ be an unbounded linear operator such that for all $\lambda>c$ ($c$ some constant), $(\lambda I-A)^{-1}$ exists and is a bounded operator which satisfies: ...
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Properties shared by equivalent norms.

I am interested in knowing about "geometric" properties shared by equivalent norms on a Banach space. Here I mean "geometric" as opposed to topological, and probably in particular with reference to ...
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52 views

Inverse of operator is not continuous in Banach spaces

Let $X$ be a Banach space. If $A:X\to X$ is an invertible bounded operator (injective, surjective and continuous), then $A^{-1}$ is also bounded. Now can I have an example of an unbounded operator ...
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42 views

If $A$ is a bounded operator on $c_0$, then $\sum_{n=1}^\infty |(Ae_n)(m)|$ is bounded uniformly in $m$

The following is Exercise 7,section 1, chapter 6 of Conway's A course in Functional Analysis. Let $A\in {\cal B}(c_0)$ (${\cal B}(c_0)$ is linear bounded operators on $c_0$) and for $n\geq 1$, define ...
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55 views

Using Stone–Weierstrass theorem for completely regular space

Let $X$ be completely regular. If $K$ is a compact subset of $X$, define $$p_K(f)=\sup\{|f(x)|;x\in X\}$$ then $\{p_k; \text{K is a compact }\}$ is a family of seminorms that makes $C(X)$ into a ...
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36 views

The restriction of an open bounded linear operator

I need some help with this question. Let $X$ be a Banach space and $T:X \to X$ be a bounded linear operator. Suppose that $T$ is open, and $X_0$ be a closed subspace of $X$. The restriction $T_0$ of ...
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26 views

Show that P is an L-projection iff $P^{*}$ is an M-projection

I have started reading "M-ideals in Banach spaces and Banach algebras", but I stuck on the first page. It says that "there is an obvious duality between L- and M- projections: P is an L-projection ...
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1answer
27 views

Paley projection is not absolutely summing

It is well known that the operator $P: H^{1}(\mathbb{T}) \to \ell_2$, given by $Pf= (\hat{f}(2^{n}))_{n \in \mathbb{N}}$ is bounded and its restriction $P_{|A(\mathbb{D})}: A(\mathbb{D}) \to \ell_2$ ...
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1answer
32 views

Finite Measure Space: Integral Closure = Bochner Integral

I can't sleep for so long time as the integral gives me headaches. I was looking for so many approaches. Now another one. Hope this works... Let $\Omega$ be a finite measure space and $E$ a Banach ...
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1answer
72 views

A special property of $\limsup$ in $\ell_1$

Let $w$ be any element in $\ell_1$, and $(w_n)$ be a bounded sequence in $\ell_1$ that converge to 0 pointwise. I want to prove ...