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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Another proof of Inverse Function theorem in $\mathbb{R}$

(Inverse Function theorem in $\mathbb{R}$) Suppose $I\subset \mathbb{R}$ is an open interval and $f:I\rightarrow\mathbb{R}$ is a differentiable function.If for all $x\in I$ is such that $f^{'}(x)\ne ...
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Explicit example of tensor norms

I can't find any example anywhere on the web where someone actually evaluates a non-trivial tensor norm. So I'm wondering about the simplest non-trivial case. Let $X$ be $\mathbb R^2$ with the ...
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fractional powers in Banach algebra [on hold]

Let $X$ be a Banach algebra. For $x\in X$ and $0< p< 1$, would $x^p\in X$? If not, under what conditions it holds?
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Correspondence between linear maps of a vector space into itself and linear maps of the dual into itself.

I was wondering about vector spaces and their dual. Specifically, in the context of finite-dimensional vector spaces, I asked myself if it is true that there is a one-to-one correspondence between the ...
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Extreme points of the unit balls of $l^\infty, C([0,1])$

Determine the extreme points of the unit balls of $l^\infty$, and $C([0,1])$ for real-valued functions, with the uniform norm. Is $C([0,1])$ the dual of a Banach space? I've found the extreme points ...
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Is $C([0,1])$ for $\mathbb{C}$ dual to any Banach Space?

I've been able to show that the extreme points of $C([0,1])$ are the continuous functions that take values on the unit circle. However, I'm not sure how to reason from here as to whether or not it is ...
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1answer
27 views

Extreme points of unit ball of Banach spaces $\ell_1$, $c_0$, $\ell_\infty$

Find extreme points of the unit balls of each Banach space, $l^1 $, $c_0$, $ l^\infty$ Can you help me with this one? For the first space, $l^1$, I thought there was no extreme point, but ...
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14 views

Let $E$ be the space $L^1(\mathbb{R})\cap L^2 (\mathbb{R})$ equipped with the norm $\|u\|_E = \|u\|_1 + \|u\|_2$.

I am trying to solve this but I got stuck. Help needed. $E$ is a Bananch space. Let $f(x) = f_1(x) + f_2(x)$ with $f_1\in L^\infty(\mathbb{R})$ and $f_2 \in L^2(\mathbb{R})$. Check that the mapping ...
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Continuity of a function in a locally convex topological space

I endow the space of bounded sequences with a locally convex topology $\tau$ such that $\tau$ is strictly finer than the product topology (the topology of pointwise convergence), $\tau_p$, and ...
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1answer
22 views

Adjoint operator on Banach space

Suppose $X$ and $Y$ are Banach spaces and $T:X\to Y$ is a bounded linear operator. Show that $T$ is an isometric isomorphism if and only if its adjoint $T^*$ is also an isometric isomorphism. Given an ...
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1answer
32 views

Bounded Operator Norm: Special Element

Given a Banach spaces $X$ and $Y$. Consider a bounded operator: $$T:X\to Y:\quad\|T\|<\infty$$ Then theres an element: $$\|Tx\|=\|T\|\cdot\|x\|\quad(x\neq0)$$ Does it always exist?
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18 views

Adjoint operator in Banach space

From Functional analysis, by Conway. I try to prove this exercise. If $ X $ and $ Y $ are Banach spaces and $ B \in \mathscr B(Y^*, X^*) $, then there is an operator $ A $ in $ \mathscr B(X,Y) $ ...
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1answer
10 views

Exercise about adjoint of densily defined operator between Banach spaces

Let $X$ and $Y$ be Banach spaces and $A:D(A)\subset X \to Y$ a densily defined linear operator. Suppose the graph of $A$ is closed. Then the follwing are equivalent: $D(A)=X$; $A$ is bounded; ...
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Does a Banach valued Cauchy series over arbitrary set converges?

Definitions: If $f$ is a function from a set $A$ (not necessarily countable) into a Banach space $V$, we say that the series of $f$ over $A$ converges if there exists an element $v \in V$ such that ...
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26 views

Is image of sum of maps closed?

Suppose $V$, $W$ are Banach spaces and $f$, $g:V \rightarrow W$ are continuous linear operators and $\operatorname{Im}f$, $\operatorname{Im}g$ are closed subspaces of $W$. Is it true that ...
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31 views

$L^p$ spaces and proper inclusion

Let $1≤p < q$. Prove that $L^p(\mathbb{R}) \subset L^q(\mathbb{R})$ and the inclusion is proper. I am unsure how to begin this or even prove it about $L^p$ spaces and Banach spaces.
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22 views

Isomorphism, Separable spaces

I am trying to show that: If there are sequences $(x_n)\subset X$ and $(y_n)\subset Y$ where $X$ and $Y$ are separable Banach spaces, such that $\overline{sp}\{x_n \mid n\in\mathbb{N}\}=X$ and ...
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1answer
22 views

Analytic sets,, effros borel structure

Let SB denote the set of closed subspaces of $C(2^\mathbb{N})$ equipped with the Effros Borel structure, and $A\subset$ SB be analytic. I am reading a proof that says $A_\sim = \{Z\in $SB $ \mid $ ...
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1answer
28 views

Closed set in Baire space

I am reading a book on Banach spaces. It introduces the Baire space $\mathcal{N}=\mathbb{N}^\mathbb{N}$ as the product of infinitely many copies of $\mathbb{N}$ with the discrete topology. We have ...
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1answer
34 views

Is the relative interior of a subspace which is not closed empty?

In a general Banach space, the relative interior of a linear subspace which is not closed is empty, why ?
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1answer
27 views

Does every closed subspace of a dual space correspond to a closed subspace of its predual?

Suppose $X$ is a Banach space with dual space $X^*$. If $Y$ is a closed subspace of $X$, then $Y^\perp=\{x^*\in X^*: x^*(y)=0 \text{ for all } y\in Y\}$ is a closed subspace in $X^*$. I am wondering ...
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1answer
19 views

Strong convergence of product of operators on a Banach space

If $\{T_n\},\{S_n\}$ are two sequences of bounded operators on a Banach space $X$, such that $\{T_n\}$ converges weakly to $T$, and $\{S_n\}$ converges strongly to $S$, does it follow that $T_nS_n\to ...
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18 views

Convergence in the weak operator topology implies uniform boundedness in the norm topology?

If $\{T_n\}$ is a sequence of bounded operators on the Banach space $X$ which converge in the weak operator topology, could someone help me see why it is uniformly bounded in the norm topology? I ...
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59 views

Categorical Banach space theory

Consider the category $\mathsf{NormVect}_1$ of normed vector spaces with short linear maps$^{\dagger}$ and the full subcategory $\mathsf{Ban}_1$ of Banach spaces with short linear maps. Both ...
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46 views

Orthogonal in inner product space

Let $(X,<.>)$ is an inner product space prove that $x$ and $y$ are orthogonal if and only if $||x+αy|| \ge ||x||$ for any scalar $α$ . The first direction if $x$ and $y$ are orthogonal ...
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Prove or disprove that T:[0,2π] -> [0,2π] given by Tx = sin(2014x) is a contraction

i know that if we assume $T:[a,b] \to [a,b] $ and if $|T'(x)| ≤ α \space \forall \space a≤x≤b$ then T is a contraction . but unsure of how to apply that to this question
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1answer
57 views

When is the dual ball of $L_1(\mu)$ weak*-sequentially compact?

Where could I find a direct proof showing that the dual ball of $L_1(\mu)$ is weak*-sequentially compact? Since $(L_1(\mu))^*=L_\infty(\mu)$, I mean the unit ball $B_{L_\infty(\mu)}$ of ...
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57 views

Banach space it isn't Hilbert space [duplicate]

How can give me two or three examples about Banach spaces which it is not Hilbert spaces with proof ( I mean why it isn't Hilbert spaces ) ?
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28 views

Coanalytic families of Banach spaces

Is it true that if $G$ is a coanalytic family of separable Banach spaces, which is not Borel, and $H\subset G$ is not Borel, then H is coanalytic? This is something I have come across. I am reading a ...
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Tensor norm for matrix algebra over an $L^p$ operator algebra

This is a question about whether a certain tensor norm has a certain property. The setting is that of $L^p$ operator algebras (i.e. norm-closed subalgebras of $L^p(X,\mu)$ for some measure space ...
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21 views

if $f$ is in Banach space, then $\nabla f $ is in the dual space?

I am not very deep in advanced real analysis. Could you help me decipher the following two phrases hold? 1) if $f$ is in Banach space $\mathcal{B}$, then $\nabla f $ is in the dual space ...
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27 views

Krein Milman Property

If a closed bounded (not compact) set $X$ in a Banach space $B$ (like $L^1$) has extreme point(s), must the max of a linear functional defined on $X$ occur at one of them? I suppose it depends on $B$. ...
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Multiplicative linear functionals on subalgebras

If $A$ is a commutative $C^\ast$ algebra and $C$ is a $C^\ast$ sub algebra of $A$ is it true that the characters on $C$ are just restrictions of characters on $A$. The reason I am asking this because ...
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$d(x,L)=\max\{f(x) \,| \, f\in L^{\perp},\, \|f\|=1\}$

Let $X$ be a normed space and $L$ its subspace. Let $L^{\perp}$ be a set of all functional of whose kernel contains $L$. Then $d(x_0,L)=\max\{f(x_0) \,| \, f\in L^{\perp},\, \|f\|=1\}$ I read a ...
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the second derivative for a composite function

Let $f: U\rightarrow \mathbb{R}$ where $U\subseteq X$ a normed vector space and suppose $f(\mathbf{u})\in C^{2}$.Let $\mathbf{x}\in U$ and let $r>0$ be such that $B(\mathbf{x},r)\subseteq U. $ ...
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37 views

How to prove a space is a dual space?

How does one go about proving that a space is a dual space? The only thing I can think of is to prove that the space is isomorphic to a dual space. Is there a better way to do this? Thank you
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A paradox derived from the open mapping theorem

The problem comes from Erwin Kreyszig's Introductory Functional Analysis with Applications, section 7.4, problem 4: Let $T:l^2\mapsto l^2$ be defined by $y=Tx, x=(\xi_j), y=(\eta_j), ...
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Proving completeness of $L^p$

I want to make sure my understanding of the proof is correct. For a Cauchy sequence $\{f_n\}$ in $L^p$, we want to find a $f\in L^p$ such that $f_n\stackrel{L^p}\to f$ Now, skipping the ...
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1answer
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A question about regulated functions

Let $X$ be a Banach space and consider the following definition. Definition. $f:[a,b]\to X$ is regulated if it has one-sided limits at every point of $[a,b]$, i.e. for every $c\in [a,b)$ there is a ...
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Are “most” operators on an infinite-dimensional complex Banach space “diagonalizable”? [closed]

This is true for finite-dimensional spaces, of course. To be precise, let $T$ be an operator on a complex Banach space $X$ which is not finite-dimensional. For each $\lambda \in \mathbb{C}$, let ...
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Isomorphism on dense subset

I am wondering if the following could be done. I want to show two Banach spaces $X$ and $Y$ are isomorphic. If $A$ is dense in $X$, and $B$ is dense in $Y$, is it sufficient to show there is an ...
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Separable spaces isomorphic

I am reading a proof and it states the following without proof: Two separable Banach spaces $X$ and $Y$ are isomorphic iff there are sequences $(x_n)\subset X$ and $(y_n)\subset Y$ such that ...
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Showing that $L^2\subset L^1$ for $L^2([0,t_f])$, with $t_f$ a fixed positive number.

I saw demonstrations using the Cauchy-Schwartz Inequality but I am still not convinced because the Inequality is as follows : $$ \left |\langle f,g\rangle\right | \leq \left \|f \right \|_{L_2} . ...
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Examples of Sobolev Spaces

I would like to apologize in advance for this trivial question! Does constant functions $u \equiv C$ and, in partucular, $u \equiv 0$ belong to $W_{0}^{1,2}(\Omega)$? Update 2: $\Omega \subset ...
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58 views

closed, convex, absorbing subset of a banach space

There is a nice theorem that every closed, convex, absorbing subset of a banach space includes an open ball arround $0$. Can you give an example where the theorem fails if we do not assume the subset ...
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1answer
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spectral theory expandable to arbitrary polynomials?

Given a Banach space $X$ and closed operators $A_i$ ($i \in \left\{0,...,n\right\}$) which have a common domain $D$ that is dense in $X$. An obvious candidate for the title of "generalised resolvent ...
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166 views

Does Continuity in Weak Operator Topology imply Continuity in Strong Operator Topology?

This homework problem has puzzled me for almost a year. As nobody in the class has figured it out, I would like to seek a proof or a disproof here. Problem (Prove or Disprove) Let $X$ be a Banach ...
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Are continuous bounded functions a subspace of $L^2$?

I have a problem where I need to work with functions that are square-integrable, bounded and continuous, i.e. the space $ L^2 \supset X = \left\{ f \in L^2 \mid f \text{ bounded, continuous}\right\} ...
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(Riesz's lemma) A closed subspace of a Banach space

Let V be a Banach space over R Let W be a proper closed subspace of V Prove : For any $\epsilon > 0$, there is a v $\in$_V_ such that ||v||=1 and ||v+w||$\geq$ $1$ - $\epsilon$ And my proof ...
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Banach space and its closed subspace. a vectors satisfying inequality.

V=a Banach space over R W=a proper closed subspace of V Prove : For any $\epsilon > 0$, there is a v $\in$_V_ such that ||v||=1 and ||v+w||$\geq$ $1$ - $\epsilon$ I have shown that there exists ...