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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Weak sequential completeness

It is obvious that reflexive spaces are weakly sequentially complete. Can we have a kind of a converse to this fact? Is there a non-reflexive Banach space $X$ such that both $X$ and $X^*$ are weakly ...
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74 views

Is the space $\ell^2_1$ injective? ($\ell^2_1$ = 2 dimensional(complex) space with 1-norm)

A Banach space $Z$ is said to be injective if for for any bounded linear map $\varphi: X \rightarrow Z$ and for any Banach space $Y$ containing $X$ as a closed subspace, there exists a bounded ...
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1answer
271 views

A distance-minimizing continuous projection onto a finite-dimensional subspace?

Let $E$ be a Banach space, which need not be a Hilbert space, and let $F$ be a finite-dimensional subspace of $E$. Suppose that for all $x \in E$, there exists a $y \in F$ realizing the minimal ...
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82 views

Defining multiplication and differentiation in a dual space

Let $B$ be a space of complex-valued analytic functions from some compact subset $C$ of $\mathbb{C}^n.$ $B$ is a Banach space with the uniform norm. Now, the paper I am reading says that we would ...
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92 views

Do I have a Banach space given the following norm?

This is very very similar to my other question asked three months ago. That time there was no Banach space because an integral in the norm definition allowed a counter-example. Once again I have a ...
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1answer
189 views

What is a decreasing scale of Banach spaces?

I am a having a hard time understanding a part of an article I am reading. The screen-cap is below. Basically, it's the line labeled (6) that I do not understand. I am not familiar with the circular ...
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138 views

Banach space integral via defining it in $X^{**}$ and then proving it's in $X$

Vector-valued integration is something I generally try not to think about very much. I have the impression that it can be a sort of "rabbit hole" of a subtlety if one allows it to be. So, I tend to ...
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1answer
162 views

Bounded operator and dense sets

Let $A : E \rightarrow E$ be a linear operator, and $E$ a (typically infinite-dimensional) Banach space. Suppose that $S$ is a collection of norm 1 elements of $E$ spanning a dense subspace of $E$. ...
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76 views

A basic question on the linear operators on Banach space

Given that $\cal{X}$ and $\cal{Y}$ are Banach spaces. $A:\cal{X} \to \cal{Y}$ and $Y:\cal{Y}\to \cal{X}$ are linear bounded operators. If $\rho(I_{\cal{X}} - YA)<1$, $\rho(I_{\cal{Y}} - ...
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1answer
26 views

Why if $\rho(I_{\mathfrak{X}} - YA)<1$ then $YA$ is invertible on the $R(YA)$?

I am reading an article where I am stucked at one point. Below is my problem. Given that $\mathfrak{X}$ and $\mathfrak{Y}$ are Banach spaces. $A:\mathfrak{X} \to \mathfrak{Y}$ and ...
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82 views

Distance of a point from a subspace vs. diameter

Let X = $(\Bbb R^N, \|\cdot\|)$ be a Banach space. Let $x_0 \in S^{N-1} = \{x \in \Bbb R^N : \sqrt{x_1^2+...+x_n^2}=1\}$. Denote $B^N_2 = \{x \in \Bbb R^N : \sqrt{x_1^2+...+x_N^2} \le 1 \}$. Define ...
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115 views

Need help with one basic result of linear operators on Banach spaces

$X$ and $Y$ are Banach spaces and suppose that $T$ be a closed subspace of $X$. $A:X \to Y$, $B:Y\to X$ and $X_{0}:Y\to X$ are linear bounded operators. It is given that the operator $BA$ is ...
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729 views

Uncountable basis and separability

We know that a Hilbert space is separable if and only if it has a countable orthonormal basis. What I want to ask is If a Hilbert space has an uncountable orthonormal basis, does it mean that it is ...
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236 views

Show that ($\ell^1$, $\|\cdot\|_1$) is complete

Show that the vector space $\ell^1 : = \{(a_n) : \sum_n|a_n| < \infty\}$ with the norm $\|(a_n)\|_1 : = \sum_n|a_n|$ where $(a_n)$ are sequences in $\mathbb C$ is complete. Thanks in advance.
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84 views

Distance from a point to a plane in normed spaces

How can we calculate the distance from a point to a plane in normed spaces ? where we are not inner product, for example: Calculate the distance in $(C[0,1],||\cdot||)$ endowed with the supremum norm ...
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53 views

If A is a balanced subset $\Longrightarrow$ conv$(A)$ is balanced

If $A$ is a balanced subset of a vector space $V$ then conv$(A)$ is balanced. Proof: Let $C=\{ax+by:a,b\ge0,a+b=1,x,y\in A\}$ Is enough to show that $C= $ conv$(A)$ How can we prove that $C$ ...
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57 views

If $X$ is a separable Banach space $ \Rightarrow$ $ X=\overline{\cup_{n=1}^{\infty} X_n}$ where $\dim X_n=n$

Let $X$ be a Banach space separable. How can we prove that there is a sequence of subspaces: $X_1\subset\ X_2 \subset \cdots \subset \ X_n \subset \cdots $ of $X$ such that $\displaystyle ...
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1answer
91 views

Characterization of Hilbert spaces [duplicate]

Let $X$ be a Banach space for which there exists a constant $\beta<\infty$ such that for every finite-dimensional subspace $B$ of $X$ , $d(B,\ell_2^n)\le\beta$ (where $\dim B=n$). Then $X$ is ...
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148 views

Show that the infimum of $T(v)=\|v-u\|_\infty$ is not attained

Let $X=\{u\in C([0,1]):\ u(0)=0\}$ with the norm $\|u\|=\|u\|_\infty$. Let $f\in X^\star$ be defined by $$\langle f,u\rangle=\int_0^1u(x)dx $$ Let $M=\{u\in X:\ \langle f,u\rangle=0\}$ and consider ...
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452 views

The Banach-Mazur distance is not reached

Let $X,Y$ be isomorphic Banach spaces. The Banach-Mazur distance: $$ d(X,Y)=\inf\{\|T\| \cdot \|T^{-1}\|: T:X\longrightarrow Y \ \text{is an isomorphism} \}$$ can be rewritten as: $$ ...
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98 views

geodesic metric

I'm trying to prove that the line segment is the minimizer of the distance $$d(x,y)=\inf l(\gamma),$$ where $x,y\in X$, $X$ is a Banach space, $\gamma$ is a path from $x$ to $y$ and ...
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70 views

Banach space :space of all adapted processes continuous equipped wih specific norm is complete

Let $\mathbb{B}$ be space of all adapted processes continuous equipped with the norm $\lVert Y\rVert_{\mathbb{B}}^2=E\left[\sup_{t\in [0,T]} |Y_{t}|^{2}\right] < \infty $, ...
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Don't understand this proof of equivalence of weak solutions to PDE

I'm trying to understand the proof that (c) implies (a) here in the following proposition (here, $\mathcal{V} = L^2(0,T;V)$). See the very last line in the image for that part: $$$$ $$$$ I give here ...
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183 views

Do maps between topological spaces somehow induce maps between Banach spaces?

If $X,Y$ are topological spaces and $h:X\rightarrow Y$ is a continuous map, is there some sort of induced map \begin{align*} h':C_b(X)\rightarrow C_b(Y) \end{align*} (or in the other direction) where ...
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223 views

annihilator of an intersection in infinite dimension

Given two subspaces of an infinite dimensional Banach space, is the sum of their annihilators dense in the annihilator of their intersection?
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1k views

Prove the boundedness of a bilinear continuous mapping.

Let $X,Y,Z$ are Banach spaces and $$B:X\times Y\to Z$$ is bilinear and continuous. Prove that there exists $M<\infty$ such that $$\lVert B(x,y)\rVert \leq M\lVert x\rVert\lVert y\rVert.$$ Is ...
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extreme points of the unit ball of the Schatten classes?

Suppose $1<p<\infty$. What are the extreme points of the unit ball of the Schatten classes $S^p$? See below for the definition of $S^p$: http://en.wikipedia.org/wiki/Schatten_norm
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Weakly compact implies bounded in norm [duplicate]

The weak topology on a normed vector space $X$ is the weakest topology making every bounded linear functionals $x^*\in X^*$ continuous. If a subset $C$ of $X$ is compact for the weak topology, then ...
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607 views

Strong convergence of operators

I'm working through the functional analysis book by Milman, Eidelman, and Tsolomitis, and I have a question. The book states a lemma that I'm a bit confused about: A sequence of operators $T_n\in ...
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116 views

If the relative norm and waek topologies of $B_X$ conincide at a point then the relative weak-$^\ast$ and norm topologies coincide in $B_{X^{**}}$

Let $X$ be a Banach space and let the relative norm and weak topologies coincide at a point $x\in B_X$. That means that every open neighborhood of $x$ in the norm topology contains an open ...
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28 views

A set, which appropriately scaled is expressible as sums of elements of a compact set is pre-compact

Assume $X$ is a Banach space and $K\subseteq X$ is compact. Let $C\subseteq X$ be such that $(\forall x\in C)(\exists x_1,x_2\in K)(2x=x_1+x_2)$ Does it follow that $C$ is pre-compact? In particular ...
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164 views

approximation property

In I. Namioka and R. R. Phelps's your paper "Tensor products of compact convex sets" Pacific Journal of Mathematics, Vol. 31, No. 2, 1969), they gave the following definition of approximation ...
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1answer
45 views

$(\cdot,\cdot)$ in Banach spaces?

I have been doing some research on fixed point theorems, and they have brought me around to papers from the 1960s in functional analysis in Banach spaces. I think that today it is common practice to ...
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101 views

Calculating the norm of $S\otimes T$ for bounded linear $S,T$.

Let $X,Y,W,Z$ be Banach Spaces. Let $X\otimes_{\pi}Y$ denote the tensor product endowed with the projective norm. If I have $S\in B(X,Z)$, $T\in B(Y,W)$, it is straightforward to show that $S\otimes ...
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Proving that Tensor Product is Associative

I want to show that $X\otimes(Y\otimes Z)$ is isomorphic to $(X\otimes Y)\otimes Z$. Intuitively I think I should just choose bases $\{e_{i}\}_{i\in I}, \{f_{j}\}_{j\in J}$, and $\{g_{k}\}_{k\in K}$ ...
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1answer
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Doubt in the proof that $l^{p}$ is complete

I was looking at the proof that $l^{p}$ is complete with respect to the standard metric. Suppose $x^{(n)}$ is a Cauchy sequence in $l^{p}$. Then Given $\epsilon > 0$, $\exists\,\, n_{0} \in ...
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262 views

About functions of bounded variation

I got the following the following idea in one of the articles that I'm reading. It goes this way. Let $X$ be a Hausdorff topological vector space and let $\mathcal{D}$ be the family of all divisions ...
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50 views

The set of finite “variations” of an unconditionally convergent series is pre-compact

Proposition: If $\sum x_i$ is an unconditionally convergent series in a Banach space $X$, then $S=\{\sum_{i=1}^n \varepsilon_ix_i:n\in\mathbb N, \varepsilon_i=\pm1\}$ is pre-compact. Proof: 1) ...
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37 views

The set of “variations” of an unconditionally convergent series is compact

Proposition: If $\sum x_i$ is an unconditionally convergent series in a Banach space $X$, then $S=\{\sum \varepsilon_ix_i:\varepsilon_i=\pm1\}$ is compact. Proof: 1) $\{-1,1\}^{\mathbb N}$ is ...
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Questions about $B(H)$ and $B(H)/K(H)$ as Banach space

I am trying to investigate the relation between Uniformly Convexity and existence of Schauder Basis for a Banach space. I read in a Handbook article that $B(H)$ (the algebra of all bounded operators ...
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$\operatorname{sconv}(A)=\left\{ \sum_{i=1}^\infty\lambda_i\cdot a_i:a_i\in A, \lambda_i\ge0,\sum_{i=1}^\infty\lambda_i=1\right\}$ is superconvex

Let $X$ be a Banach space and $A\subset X$ a subset bounded. Denote by $\operatorname{sconv}(A)$ the superconvex hull of $A$: $$\operatorname{sconv}(A)=\left\{ \sum_{i=1}^{\infty}\lambda_i\cdot ...
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129 views

Invariant subspace associated with a complex eigenvalue

Let $E$ be a Banach space over the reals and $T:E\to E$ a linear operator. Suppose that $\mu=re^{i\theta}$ is a eigenvalue of $T$ with $r>0$. How can one find two vector $z_1,z_2\in E$ such that if ...
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Counterexample for the stability of orthogonal projections

Let $V$ be a seperable Banach space, which is dense and continuously embedded in a Hilbert Space $H$. Let $(V_m)$ be a Galerkin scheme (See definition below) for $V$. Using the embedding we can ...
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68 views

application of the theorem of the open application

Let $ X, Y $ be Banach spaces. Suppose that $ T: X \to Y $ is a compact operator. show that if $ \dim Y $ is infinite, then $ T $ is not surjective. idea: Using the theorem of the open application
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A basic question on Type and Cotype theory

I'm studying basic theory of type and cotype of banach spaces, and I have a simple question. I'm using the definition using averages. All Banach spaces have type 1, that was easy to prove, using the ...
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1answer
28 views

$K(t,x)=\inf_{x=a+b,\ a\in X,\ b\in Y}\{\|a\|_X+t\|b\|_Y\}$ and $\int_0^\infty \frac{|K(t,x)|^p}{t}dt<\infty$ implies $x=0$?

Let $X,Y$ be two Banach spaces with respective norms $\|\cdot\|_X$ and $\|\cdot\|_Y$. Suppose that $X$ and $Y$ are subsets of a vector space $Z$. Define $K(t,x)$ for $t\in (0,\infty)$ and $x\in X+Y$ ...
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59 views

Family of complemented subspaces

Let $X$, $Y$, $A$, $B$ be topological vector spaces. Given two jointly continuous families of linear injective maps $P: Y \times A \rightarrow X$ and $R: Y \times B \rightarrow X$, such that for $y=0$ ...
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416 views

If $C$ is convex , weakly-closed and norm-bounded $\Longrightarrow$ $C$ is weakly-compact

Let $X$ be a Banach space and $C\subset X$. $\fbox{1}$ If $C$ is convex , weakly-closed and norm-bounded $\Longrightarrow$ $C$ is weakly-compact ? $\fbox{2}$ If $C$ is convex , weakly-closed ...
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1answer
94 views

$K$ is weakly-compact $\Longleftrightarrow$ $\Pi(K)$ is weak*-compact

Let $X$ be a Banach space and $K\subset X$. $\displaystyle \Pi:X \longrightarrow X$** canonical injection $\Pi(x)(f)=f(x)$ How can we prove that: $K$ is weakly-compact $\Longleftrightarrow$ ...
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1answer
115 views

The topology on bounded sets in $X$** of pointwise convergence on $B$ is metrizable

Let $X$ be a Banach space. If $B\subset X$* is a norm-separable How can we prove that: The topology on bounded sets in $X$** of pointwise convergence on $B$ is metrizable. $X$*$=B(X,\mathbb{R})$ : ...