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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$B$ be an uncountable dimensional real Banach space and $T:B \to B$ be a surjective isometry such that $T(0)=0$ , then is $T$ linear?

Let $B$ be a real Banach space and $T:B \to B$ be a surjective isometry such that $T(0)=0$ , then is $T$ linear ? I know that it is true if $B$ is a Hilbert space ( we only need an inner product ...
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In $(C[0,1],||.||_{\infty})$ , is the set $\{p(x)\in C[0,1]$, $p(x)$ is a polynomial $:\int_0^1 p(x)dx=1\}$ totally bounded?

Consider the metric space $(C[0,1],||.||_{\infty})$ , in this space , is the set $\{p(x)\in C[0,1]$, $p(x)$ is a polynomial $:\int_0^1 p(x)dx=1\}$ totally bounded ? Please help , Thanks in advance
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subspace of approximation property

A Banach space $X$ has the approximation property (AP) if for any compact subset $K$ of $X$ and any $\epsilon>0$, there exists a bounded finite rank operator $R$ such that $\| x - R(x) \| < \...
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Proving continuity of a operator $T\colon E \to E'$ [duplicate]

Let be $E$ a Banach space over real numbers and $T\colon E \to E'$ linear such $T(x)(x)\geq 0$ for all $x\in E$, prove T is continuous. If $x_n\to x$ and $T(x_n)\to \phi\in E'$ then $T(x_n)(y)\to\...
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Norms on an Ultraproduct

Suppose $X$ is a Banach space and $\mathcal{U}$ is a non-principal ultrafilter on $\mathbb{N}$. I am interested in the Banach space $(X)_\mathcal{U}$, where we consider sequences $(x_i)_{i \in \mathbb{...
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Space of Lipschitz continuous functions is complete

Let $X$ be set of functions $f:[-1,1]\to \mathbb{C}$ such that $f(0)=0$ and there exists $\alpha>0$ such that $$ |f(t)-f(s)|\le \alpha |t-s| $$ for all $t,s\in [-1,1]$. Equip $X$ with the norm: $...
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22 views

Bounded linear extension of an operator whose co-domain is incomplete.

Let $X$ be a normed space and $Y$ be a Banach space. The bounded linear extension theorem states that a bounded linear operator $T : M \to Y$, where $M \subseteq X$ is dense, can be extended to a ...
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22 views

dom(A) is a Banach space w.r.t. the Graph-norm

Let $X$ and $Y$ be Banach spaces and let $A:dom(A)\to Y$ be a linear operator, defined on a linear subspace $dom(A)\subset X $. Prof that the graph of $A$ is a closed subspace of $X\times Y$ if and ...
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21 views

A singleton as domain sum of a series

Consider the series $$e_1+\frac{1}{2}e_2-\frac{1}{2}e_2+\frac{1}{4}e_3-\frac{1}{4}e_3+\frac{1}{4}e_3-\frac{1}{4}e_3+\frac{1}{8}e_4-\frac{1}{8}e_4+\cdots-\frac{1}{8}e_4+\frac{1}{16}e_5-\cdots$$ in the ...
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64 views

Trouble finding a function satisfying an integral equation

I'm stuck at the last step of this exercise: b) Use the Banach fixed point theorem to show that there is a unique function $f \in C[0,1]$ for which the equation $$f(t) + \int_0^1e^{\tau+t-3}f(\tau)...
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Is it true that the closed convex hull of a compact subset of the dual equipped with the w*-topology is compact?

Let $X$ be a Banach space. Consider the dual $X^*$ equipped with the weak*-topology. Is it true that the closed convex hull of a compact subset $K$ of the dual $X^*$ is compact? ps: I know that the ...
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extending a functional while preserving its norm

Suppose $E$ is a finite-dimensional subspace of a Banach space $X$, and $x_0\in X$ is a vector with $x_0\notin E$. Suppose $f\in X^*$ is a continuous linear functional and that $a>0$ is a positive ...
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48 views

How to find out if a function belongs to $H^2$ or $H^1$

I'm beginning with Sobolev spaces and I found out, that $$ H^k = W^{k,2}. $$ I've also seen the following exercise recently: $$ \frac{1}{2}u'' = 1 $$ And here I'm supposed to find out if $u$ ...
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55 views

When is $C(X)$ reflexive?

Let $X$ be a compact Hausdorff space. What are sufficient and necessary conditions on $X$ under which $C(X)$ would be a reflexive Banach space. Is there a non reflexive Banach space $C(X)$ such that $...
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Analyticity of a certain Banach manifold

I am trying to make sense of some infinite dimensional topology. Let $\mathcal{A}_r$ to be the set of all real-analytic functions $\mathbb{C}^2\to\mathbb{C}^2$, whose radius of convergence is at ...
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Bijection from linear subspaces extends to their closures

Let $E,F$ is Banach spaces, and $\pi : E \to F$ is linear bounded surjective operator. Let $N$ is linear subspace (not necessary closed!) of $E$, $M$ is linear subspace (not necessary closed too) of $...
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Showing that $(a_n)_n \in l_1$ provided $\sum_{k=1}^\infty a_kx_k$ exists for any $(x_n)_n \in c_0$

I tried first using the fact that $c_0$ is Banach to apply the Uniform Boundedness Principle on the function series $(T_n)_n = \{\sum_{k=1}^n a_kx_k\}$, $T_n:c_0 \rightarrow \mathbb{K}$, and then to ...
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36 views

spectrum of an operator restricted to an invariant subspace

Let $X$ be an infinite-dimensional real Banach space and $T\in\mathcal{L}(X)$ a continuous linear operator acting on $X$. Suppose $W$ is a finite-codimensional $T$-invariant closed subspace of $X$, ...
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Every Cauchy sequence in $\{f\in (C([0,1]),\|\cdot\|_1)\,|\,\exists a,b\in\mathbb R:f(x)=ax+b\}$ converges

I have trouble proving that, using the norm $\|f\|=\int_0^1|f(x)|\mathrm dx$, for a Cauchy sequence of functions $f_n(x)=a_nx+b_n$, the sequences $(a_n)_n$ and $(b_n)_n$ also have to be Cauchy ...
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The set of w*-continuous operators is closed for the weak* topology?

Let $X$ be a dual Banach space, i.e. $X=(X_*)^*$ for some Banach space $X_*$. Consider the weak* topology of $B(X)$, i.e. the topology of pointwise convergence on $X$ endowed with the $\sigma(X,X_*)$-...
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A problem of nets in topology

Let $G$ be a topological group with neutral element $e$. Let $\pi \colon G \to B(E)$ a (non-continuous) representation of $G$ on a Banach space $E$ by bounded linear operators. Let $T$ an element of ...
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Vectors in Normed Space Must Have Finite Length?

I have assumed this to be the case, and consequently this is why one looks at convergent sequences of vectors in normed, Banach, and Hilbert spaces. But, I've never seen this listed explicitly as an ...
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Bijectivity and Lipschitz continuity of a function on a Banach space

I don't really know how to solve the following exercise, I need a little help: a) Let $(X,\|\cdot \|)$ be a Banach space and $F\colon X \to X$ Lipschitz-continuous (i.e. $|F(x) - F(y)| \le L|x-y|$)...
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Applying a functional over an infinite sum

The space $\ell^p$ has a Schauder basis and so we can uniquely express any element of $\ell^p$ as: $$x= \sum^\infty_{i=1}{\alpha_i}{e_i}$$ Where $(e_i)$ is the Schauder basis for $\ell^p$. My ...
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Peano's theorem , initial value problem, Banach Spaces

I'm Taking a Course in Differential Equations and this is one of the exercises those I have to do at home, I can't come up with these short questions: Let X be an infinite-dimensional Banach space. ...
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Let $X=\left\{ f:[0,1]\to\mathbb R \;\big|\; \lVert f\rVert_X<\infty,\ f(0)=0\right\}$. Show $(X,\lVert\cdot\rVert_X)$ is complete.

The following is a problem on an old Analysis preliminary exam at my institution; I'm prepping for the prelim. The problem is: Let $$X=\left\{ f:[0,1]\to\mathbb R \;\big|\; \lVert f\rVert_X<\infty,...
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21 views

Does the norm of an operator on a $w^{*}$ dense subspace determine its norm?

Let $X$ be a (separable) Banach space, $T:X^{*}\to X^{*}$ a bounded operator, and $Y\subset X^{*}$ a norm closed, $w^{*}$ dense subspace of $X^{*}$. Is it true that $\|T\|=\|T_{|Y}\|$?
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Is there a Banach space which is not isometrically isomorphic to $l^p$?

I know that every Hilbert space is isometrically isomorphic to $l^2(\beta)$ where $\beta$ is a Hilbert basis for that space. Do Banach spaces have the similar property? That is, is every Banach space ...
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If $\varphi \in E'$ and $A$ is convex and open then $\varphi (A)$ is an open interval

Let $E$ be a real normed space and $\varphi \in E'$, $\varphi \neq 0$. Suppose that $A \subset E$ is an open convex not empty subset. Show that $\varphi(A)$ is an open interval. Since $A$ is ...
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28 views

Closable Operators: Nonexample

Given the Banach space $X:=\mathcal{C}([0,1]\cup[2,3])$. I remember I've seen a beautiful example of a non-closable operator whose graph is dense. It involved exploiting Stone-Weierstraß for a ...
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22 views

Hahn-Banach: Operators

Given two Banach spaces $X$ and $Y$. (More generally locally convex spaces) Regard a closed subspace $U\subseteq X$. Does every bounded operator extend: $$T\in\mathcal{B}(U,Y)\implies T_E\in\mathcal{...
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35 views

On Schauder basic systems in universal enveloped algebra of system of countable family of bounded selfadjoint operators

Let $A = C^*(1,T_1,T_2, ... | T_i^* = T_i, ||T_i|| \leqslant 1)$ - universal enveloped $C^*$-algebra of countable family of selfadjoint operators. I want to know as more as possible about that algebra,...
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If $A$ is bounded and dissipative, is $\lambda \mathbb 1-A$ invertible for $\lambda>0$?

Let $X$ be a Banach space and $j$ a map (not necessarily linear or anti-linear!) from $X$ to $X'$ so that $$j_x(x)=\|x\|^2 \qquad \forall x \in X \text{ and }\qquad \|j_x\|_{X'}=\|x\|$$ We say that ...
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Proof that $C(K)$ is a Grothendieck space for $K$ an extremely disconnected compact space.

I am looking for a proof, other than the original article by Grothendieck which is in French, that the space $C(K)$ is Grothendieck when $K$ is extremely disconnected.
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Formal series which uniformly bounded in each representation of universal $C^*$ algebra converge

Let $A = C^*(T_1,T_2,...|T_i^* = T_i, ||T_i||\leqslant 1)$ - universal $C^*$ algebra of countable family of selfadjoint operators. I have formal series $x = \sum_{i_1,...,i_k} \alpha_{i_1, ..., i_k} ...
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29 views

Quotient maps in Banach spaces

I came across two definitions of quotient maps in Banach spaces. A bounded linear transformation $T:X\to Y$ is a quotient map if: A) $T(\textrm{int}(B_X))=\textrm{int}(B_Y)$ B) $\overline{T(B_X)}=...
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Unit ball of $L^1$, $L^\infty$ and $C(X)$ is not strictly convex

I need to show that the unit balls of $L^1(\mu)$, $L^\infty(\mu)$ and $C(X)$ are not strictly convex. I have already shown that if $1<p<\infty$ then the unit ball of $L^p(\mu)$ is strictly ...
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Weak convergence of Banach space valued random variables

In the book 'Probability in Banach Spaces: Isoperimetry and Processes', available here http://michel.talagrand.net/, in the second chapter on the page 34, above the Theorem 2.1 there is a statement. (...
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The quotient norm on $\ell^{\infty}(\mathbb{N}) / c_0 (\mathbb{N})$ is given by $\limsup_{n \in \mathbb{N}} |x_n|$.

I try to show that the norm on the quotient space $\ell^{\infty}(\mathbb{N}) / c_0 (\mathbb{N})$ is given by $\limsup_{n \in \mathbb{N}} |x_n|$, where $x = (x_n)_{n \in \mathbb{N}} \in \ell^{\infty} (\...
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Exercise 1.65 of Megginson's “An Introduction to Banach Space Theory”.

Unfortunately I do not succeed in completing the following exercise: Let $X$ be a Banach space and let $T : X \to \ell^{1} (\mathbb{N})$ be a linear operator. For each $n \in \mathbb{N}$, let $(Tx)...
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The Banach Mazur distance between n-dimensional space and $ \ell_{\infty} ^n$

Let $X$ be an $n$-dimensinal space. Is the Banach-Mazur distance $d(X,\ell_{\infty}^n)$ less than or equal to $n$? Is $d(X,\ell_{\infty}^n)$ less than or equal to some constant $C(n)$ depending only ...
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Exercise 1.64 of Megginson's “An Introduction to Banach Space Theory”.

Could someone verify whether my solution to the following exercise is correct? The reason I am a bit in doubt is because the chapter of which this Exercise is part consists of the Banach-Steinhaus ...
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45 views

About сonvergence of partial sums of basis of Banach space

Let $B$ is some separable Banach space with Schauder basis $\{e_i\}_{i=1}^\infty \subset B$. Let $\{\alpha_i\}_{i=1}^\infty$ - is some sequence of complex numbers. Let $p_n = \sum_{i=1}^n \alpha_i e_i$...
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Every bounded sequence of dual space contains a subsequence which is weak* convergent

I were doing this problem in Functional Analysis of Erwin Kreyszig(part 4.9, problem 10, page 269), but got stuck in the last point to come to the conclusion. Can anyone give me some hint to move on? ...
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38 views

Convergence of partial sums of basis vectors in banach space

Let $B$ is some separable Banach space with Schauder basis $\{e_i\}_{i=1}^\infty \subset B$. Let $\{\alpha_i\}_{i=1}^\infty$ - is some sequence of complex numbers. Let $p_n = \sum_{i=1}^n \alpha_i e_i$...
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34 views

Uniqueness of endpoints of half-open line segments in linear spaces.

I try to solve the following exercise, which is Exercise 1.18 in Robert Megginson's An Introduction to Banach Space Theory. Let $X$ be a linear space, and define for any $x_1, x_2 \in X$ the line ...
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1answer
28 views

Lipschitz-continuity of $x\mapsto\frac{x}{||x||}$ in a general Banach space

Let $(X,||.||)$ be a Banach space. Assume we have constants $0<C_1<C_2<\infty$. Define the set $A:=\{x\in X\text{ }|\text{ } C_1\le ||x||\le C_2\}$. Is the map $f\colon A\rightarrow X$, $x\...
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1answer
40 views

About $L^\infty$ in the dual of $L^1$

Does someone know where can I find a proof of the following? $$L^\infty(\mathbb{T}, X^*)\subseteq\left(L^1(\mathbb{T},X)\right)^*$$ (where $X^*$ denotes the dual space of $X$).
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26 views

If a linear map $T:X^*\to X^*$ is norm-norm continuous, is it weak-star - weak-star continuous?

Let $X$ be a Banach space and suppose $T:X^*\to X^*$ is a linear mapping. If $T$ is norm-norm continuous, i.e. continuous from the normed space $X^*$ into the normed space $X^*$, is it also continuous ...
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34 views

Projective tensor product continuous?

For $V$, $W$ Banach spaces, is the canonical morphism $B(V) \times_\pi B(W) \to B(V \hat{\otimes}_\pi W)$ continuous, where $B(V):=B(V,V)$ are linear bounded endomorphisms with operator norm? If so, ...