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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Space of Functions: Characterizations of Positivity

Context The problem here is about the characterization of positivity for real or complex valued functions: $$\sigma(f)\geq 0\iff\sigma(f(x))\geq 0\text{ for all }x\in X\iff f(x)\geq 0\text{ for all ...
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Banach Spaces: Totally Bounded Subsets

As an easy consequence of Riesz' lemma it is known that infinite dimensional Banach spaces possess bounded subsets that fail to be totally bounded. On the other hand in finite dimensional Banach ...
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52 views

Sums of special vectors

Let $v$ be a vector obtained by taking a sum of $k$ vectors the of the form $(0,0,\ldots,0, -n, *,*,\ldots,*)$, where $"*"$ stands for either $0$ or $1$, and the position of the $-n$ entry can vary ...
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54 views

Banach Spaces: Totally Bounded vs. Bounded

Are the finite dimensional Banach spaces precisely those ones in which subsets are totally bounded iff they're bounded?
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completion of a normed space [on hold]

Prove that if $M$ is a normed linear space,then there exists a unique(up to isomorphism)Banach space $X$ containing $M$ such that $\overline{M}=X$ Please help me.Thanks in advance.
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When is an operator on $\ell_1$ the dual of an operator on $c_0$?

Suppose $T:\ell_1\to\ell_1$ is a continuous linear operator. When can we say that $T$ is a dual, or adjoint, of an operator on $c_0$? In other words, under what conditions can we find a continuous ...
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28 views

Bounded below adjoint operators

Let $T\colon X\to Y$ be a bounded linear operator. Suppose that $Z$ is a subspace of $Y^*$ such that $T^*$ is [bounded below][1] on $Z$. Denote by $\text{w*-dens}\, Z$ the minimal cardinality of a ...
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52 views

Discontinuous seminorm on Banach space

We have known that if $X$ is a Banach space and $\sum_{n=0}^{\infty}x_n$ is an absolutely convergent series in $X$ then $\sum_{n=0}^{\infty}x_n$ is a convergent series. Moreover, we have $$ (*)\quad ...
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61 views

Weak*-complemented subspaces of $\ell_\infty$

Consider $\ell_\infty$ as $\ell_1^*$. Let $X$ be an infinite-dimensional complemented subspace of $\ell_\infty$ (in partiuclar, $X$ is isomorphic to $\ell_\infty$). Can we find a further subspace ...
3
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1answer
56 views

Small $\ell^p$ spaces are obtainable from $L^p$

I've seen that in a lot of books there is written that $$l^p=L^p(X,\Sigma,\mu),$$ where $X=\Bbb N, \Sigma=P(\Bbb N), \mu=\#$, ($\#$ is the counting measure). I would like to see how to prove it, ...
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22 views

Does a strictly convex and weak metrizable unit sphere of a Banach space imply locally uniform convexity?

I'm trying to find a proof for this question Let $X$ be a Banach space with strictly convex norm and a metrizable unit sphere, $S_X=\{x\in X: \|x\|=1\}$, for the weak topology. Does $X$ admits ...
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40 views

Question on the proof of Open mapping Theorem

I am studying on Open Mapping Theorem. I am stuck at a point in the proof. Open Unit Ball: A bounded linear operator T from a Banach space X onto a Banach space Y has the property that the image ...
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42 views

Existence of a semigroup of bounded operators which is not $C_0$

Let $X$ be any Banach space. Then we can define a $C_0 $ semi group of bounded operators on $X$. But my question is that can we define a semi group of bounded operators which is not $C_0$?
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41 views

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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1answer
28 views

The image of non-reflexive Banach space in its bidual is at large distance from some unit vector in the bidual

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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75 views

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

Let $E_n$ be the set of all $f \in C\big([0,1]\big)$ for which there exists $x_0 \in [0,1]$ (depending on $f$) such that \begin{align*} \lvert\, f(x)-f(x_0)\rvert \leq n\lvert x-x_0\rvert, ...
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31 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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22 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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1answer
27 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 $i\colon E\to Eā€™$ be a dense, compact inclusion of $E$ into some other Banach space $Eā€™$. Finally assume that ...
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49 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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46 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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36 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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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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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 ...
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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 ...
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42 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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124 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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47 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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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 ...
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81 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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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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37 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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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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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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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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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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26 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 ...
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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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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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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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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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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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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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415 views

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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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 ...