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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Non linear compact map

Suppose to have two Banach spaces $E$ and $F$, with $E$ reflexive. Suppose to have a continuous map $T:E \to F$ which maps bounded subsets into precompact subsets. $T$ is not assumed to be linear. ...
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Existence of a basic sequence with basis constant $1$ in a Banach space

I am interested in a problem motivated by the following theorem. A proof (and the relevant definitions) can be found in many textbooks about Banach space theory, see for example Corollary 1.5.3 in ...
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264 views

In a normed space, the sum of a Closed Operator and a Bounded Operator is a Closed Operator.

The book "Introductory Functional Analysis with Applications" (Kreyszig) presents the following lemma Let $T:\mathcal{D}(T)\to Y$ be a bounded linear operator with domain $\mathcal{D}(T)\subset X$...
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239 views

Proving a particular function is surjective on a Banach space.

Let $(E,\|x\|)$ and let $f: E \to E$ such that $f+Id$ is a contraction ($Id$ is the identity map). Prove that $f$ is surjective and prove that, if $f$ is linear, then $f$ is a homeomorphism. The ...
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1answer
152 views

Absolutely convergent sums in Banach spaces

Let's say a sum of elements in a Banach space is absolutely convergent if even the sum of the norms converges, i.e. $\sum_{i=1}^\infty ||x_i|| \le \infty$. This condition implies that the sum of the $...
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Understanding a theorem from “Probability theory of Banach Spaces ” book.

I don't understand the proof after "The hypothesis of the theorem indicate...... , can someone kindly explain it for me . Thanks :)
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3answers
671 views

When do inner products of weakly convergent subsequences converge?

If we have 2 weakly convergent subsequences in $L^2(U)$ (for $U$ some bounded open domain with smooth boundary), $u_k\rightharpoonup u$ and $v_k\rightharpoonup v$, under which conditions do we have $$\...
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2answers
160 views

A problem about Linear Operator

$X$ and $Y$ are Banach Spaces.$ T$ is a linear bounded operator from $X \to Y$. There exists a real number $c$ which is positive, such that for any $y$ belonging to $T(X)$, there exists a $x$ which ...
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1answer
726 views

Dual of the space of all convergent sequences

I need to find what it wrong with my logic and Ii will be glad if someone can told me what I do wrong. Define $C$ be the subspace of $ l^{\infty} $ that consists of convergent sequences and let $C_0$...
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1answer
150 views

Compact Operators and Complete Metrics Spaces

I have a couple of questions about compact operators and compactness in complete metric spaces: 1.I have the following implications: Let $Y$ be a metric space with $A$ a subset of $Y$. $A$ is ...
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1answer
91 views

Prove that $|x(a)| + \max \{|x'(t)|: t \in [a,b]\}$ is a norm for a complete space

Prove that the space $C^1([a,b])$ consisting of continuous functions in $[a, b]$ with the norm $|x(a)| + \max \{|x'(t)|: t \in [a,b]\}$ is a Banach space. I can't prove the completeness of this ...
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108 views

Second dual of a Grothendieck space

The classical example of a Grothendieck space is $\ell_\infty$. It is also known that its even duals $\ell_\infty^{**}$, $\ell_\infty^{****}$, $\dots$ are Grothendieck spaces. (See, e.g., this paper ...
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1answer
126 views

every denting point and strongly exposed point is extreme point

If $X$ be a Banach space and $K$ is a subset of $X$, then I want to prove Every denting point of $K$ is extreme point Every strongly exposed point of $K$ is extreme point $K$ is the closed convex ...
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1answer
93 views

About measurability of operators

I'm triyng without success, to find some examples of functions that: $\bullet$Are WOT-measurable, but not SOT-measurable. $\bullet$Are SOT-measurable, but not $||\cdot||$-measurable. I give the ...
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1answer
196 views

Quotient norm and actual norm

I have a question about the proof that $X\backslash U$ is a Banachspace if $X$ is one and $U$ is closed. In my book it is said, that for $x_k \in X$ and a series $\sum_{k=1}^{\infty}||[x_k]||< \...
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1answer
60 views

Matrix-valued function

I have a problem about matrix-valued function. Given a function $f:\mathbb{R}^k \rightarrow {\cal M}_{k \times k}$ of class $C^1$, where ${\cal M}_{k \times k}$ is the set of all $k \times k$ matrices....
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1answer
435 views

Application of Uniform Boundedness Theorem to prove an equivalence involving sequences.

After state and prove the Uniform Boundedness Theorem, the Kreyszig Functional Analysis book presents the following problem: I'm trying to solve it but I need help to finish it. What I have done (...
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1answer
225 views

distance it's attained on closed and finite dimensional subspaces of a Banach space

Let $X$ be a Banach space. And let $F\subset X$ be a closed and linear subspace (in particular is Banach). I want to prove the following: Let $d(x,F)=\displaystyle \inf_{y\in F} ||x-y||$. Is it true ...
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1answer
49 views

Determine when operator is compact

Let $B$ be the Banach space of bounded complex functions on $[0,1]$ with sup-norm. For $q \in B$, define the (multiplication) operator $T_q : B\rightarrow B$ by $(M_q f)(t) = q(t)f(t)$. Which $q$ ...
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1answer
66 views

Closure of $l_p$

How can I find closure of $l_p$ in $l_\infty$? I was trying to show that the closure is the whole space but I have failed. What's more now I think that it's not true. I have no idea how to start doing ...
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2answers
106 views

Reflexive, separable containing all finite dimensional spaces almost isometrically

Is there a separable, reflexive Banach space $Z$ such that for every finite dimensional space $X$ and every $a>0$, there is a $1+a$-embedding of $X$ into $Z$? I can do the question without the '...
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1answer
284 views

Proof of the nonexistence of an identity $\phi$ involving convolution

The Banach space $L^1(\mathbb{R}^n)$ is an algebra with a product (convolution) which is both commutative and associative. But this algebra does not have a multiplicative identity. An attempt to show ...
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Let $V$ be a normed vector space over $\mathbb{C}$, is there an inner product structure on $V$ such that the two spaces have the same topology.

Let $V$ be a complete normed vector space over $\mathbb{C}$. Let $\tau_1$ be the topology induced by its norm. Is there an inner product structure on $V$ such that the topology induced by the norm of ...
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1answer
143 views

If a Banach space $X$ is isometric to its first dual $X^*$, must $X$ be reflexive?

Suppose that $X$ is a Banach space such that there exists a linear isometry $X \rightarrow X^*$. Must $X$ be reflexive? Of course, this implies that $X$ is isometric with its second dual $X^{**}$. ...
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149 views

Linear subspace of Banach space containing unit ball

Am I right that any linear subspace of Banach space which contain unit ball is whole space?
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1answer
68 views

compute the norm of a compact operator on $l^2$

Let $a_j\to 0$ and let $T:l^2 \to l^2$ be the operator defined by $ T(s_1,s_2,s_3,...)=(0,a_1s_1,a_2s_2,...)$. Compute the operator norm $||T||$. The hint of the problem is prove that $T$ is a ...
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3k views

is bounded linear operator necessarily continuous?

Let $U, V$ be separable Banach spaces. Suppose we have a bounded, linear operator $C : U\to V$. Questions are the following *) Shall $C$ be continuous since $V$ is a Banach space? *) In general, ...
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1answer
252 views

Linear functionals separate point and closed subspace of Banach space

Let $\mathcal{B}$ be a real Banach space and $S$ a closed subspace of $\mathcal{B}$. I want to prove that for $f_0 \notin S$, there is a linear functional $l$ such that $l(f) = 0$ for all $f \in S$ ...
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1answer
171 views

transpose of the exponential operator

Let $X$ be a Banach space and $T:X\to X$ be a continuous and linear operator. What is the transpose operator of $e^T?$ I would like to prove that $e^{T'}=(e^T)'$. At least that equality make perfect ...
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prove that $T''$ is not injective (difficult computation) [duplicate]

Let $T:c_0 \to c_0$ defined by $T(\{s_j\}_j)=\{s_{j+1}-s_j\}_j$. Prove that $T''$ is not injective. I tried even knowing that $(c_0)' \sim l^1$,in fact if $F\in (c_0)'$ then there exist $s=(s_j)\in l^...
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1answer
54 views

prove that this double transpose $T''$ it's not injective.

Prove that if $T''$ is injective then $T$ it's injective, but the converse it's not true. I proved the first part " $T''$ injective implies $T$ injective". It's easy to prove that $T''$ is ...
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1answer
234 views

Can a meager linear subspace be written as a countable increasing union of nowhere dense subspaces?

Let $X$ be a separable Banach space. In this question, "subspace" means a linear subspace, not necessarily closed. Suppose $E \subset X$ is a subspace which is meager, so that we can write $E = \...
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compute the spectrum of an operator

Let $C_{0}(\mathbb R)$ be the space of complex continuous valued functions on $\mathbb R$ which vanish at $\infty$ equipped with the sup-norm. Let $S:C_{0}(\mathbb R) \to C_{0}(\mathbb R)$ given by $(...
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1answer
146 views

a compact operator on $l^2$ defined by an infinite matrix

Let $A$ be an infinite matrix such that $\displaystyle \sum_{i,j}|a_{i,j}|^2<\infty$. Then $A$ defined a compact operator on $l^2$.
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275 views

prove that this operator is not compact

Let $g\in C[0,1]$ be a continuous function and $g\ne 0$. Let $G:C[0,1]\to C[0,1]$ the operator defined by: $G(f)(x)=f(x)g(x)$. I proved that the operator is linear and continuous. I want to prove that ...
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1answer
117 views

Density of linear span of idempotents in $L^{\infty}$

How do I show that the linear span of idempotents is dense in $L^{\infty}(\Omega,\mu)$ where $(\Omega,\mu)$ is a measure space? I don't really have any idea how to do this. Does it involve ...
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1answer
106 views

A completely continuous operator carries weakly Cauchy sequences into norm-convergent ones

Let $X,Y$ be Banach spaces. Show that if $T:X\to Y$ is a completely continuous operator, then $T$ carries weakly Cauchy sequences into norm-convergent sequences. Let $(x_n)_{n=1}^\infty$ be a weakly ...
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1answer
92 views

prove that the operator is compact.

Let $H$ be a Hilbert space over $\mathbb C$, and $\{f_j\}$ a orthonormal set in $H$. Let $t_j\in \mathbb C$ such that $\displaystyle \lim_{n\to \infty} t_j =0$ i.e $(t_j)_{j\in \mathbb N}\in c_0$. ...
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71 views

unconditional basis inequality

I'm reading a paper and having some trouble with a certain inequality. Let $W$, $X$, and $Y$ be Banach spaces, with $(x_n)$ a normalized basic sequence in $X$ and $(w_n)$ a normalized unconditional ...
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2answers
97 views

Multiplication operator with a function non-vanishing on the cantor set

Let $M_f$ be the multiplication operator, which acts on bounded functions $g$ on the unit interval as $g\mapsto fg$, with $f:[0,1]\rightarrow \mathbb{C}$ such that $f$ is nonzero only on the Cantor ...
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428 views

Let $E$ be a Banach space, prove that the sum of two closed subspaces is closed if one is finite dimensional

Let $E$ be a Banach space and let $S$ and $T$ be closed subspaces, with dim$\space T<\infty$. Prove that $S+T$ is closed. To prove that $S+T$ is closed I have to show that for any limit point $x$ ...
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179 views

Dual space is 1-complemented in the third dual

Given $X,Y$ be normed spaces over $\mathbb k$ (where $\mathbb k$ could be $=\mathbb C$ or $\mathbb R$). Let $T\in L(X,Y) $ (a continuous and linear operator) then we define $T':Y'\to X'$ by $T'y'=y'T$,...
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1answer
53 views

Is the domain of a closure operator close?

Let $X$ and $Y$ be two Banach spaces. Let $A:D(A)\subset X \to Y$ linear and continuous. Then $A$ is closable. Let $\overline{A}$ be the closure operator. My question is: is $D(\overline{A})$ close?
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1answer
240 views

Functional analysis, help in Hahn-Banach theorem application

$M$ is the subspace of $L_p[a,b]$ that $\forall f\in L_p[a,b]$ $\exists g\in M$ with $f(t)\leq g(t)$ almost everywhere. $T:M\rightarrow \mathbb{R}$ $\quad$$T(f)\geq0$ whennever $f(t)\geq0$ a.e ...
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1answer
70 views

Limit of a sequence that is Cauchy with respect to a family of seminorms

Definition: Let $p$ be a seminorm on some space $X$. A sequence $(x_n)$ in $X$ is said to be Cauchy with respect to $p$ if for any $\epsilon > 0$, there is an $N > 0$ such that $p(x_m - x_n) <...
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499 views

Closure of the range of a compact operator

Let $X$ be an infinite-dimensional Banach space, and let $Y$ be a banach. Let $T$ be a compact operator from $X$ to $Y$, ie. if $(x_n)$ is a sequence in $X$ then there is a subsequence s.t. $T(x_{n(k)}...
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347 views

Every Banach space isomorphic to a subspace of C(X) (for some X)

The book of Douglas says on page 12: Theorem (Banach): Every Banach space $B$ is isometrically isomorphic to a closed subspace of $C(X)$ for some compact Hausdorff space X. Proof: Let X be $(B^*)_1$ ...
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202 views

Weaker Than The Weak Topology?

The weak topology on a Banach Space $E$ is defined to have sub-base consisting of open balls of the form $B_\alpha(x,r) = \lbrace y \in E : \vert \alpha(x-y) \vert < r\rbrace $ for each $x \in E$ ...
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1answer
251 views

Subspace isomorphic to a complemented subspace

I'll begin by writing down the definitions I'm using, to avoid confusion. Let $X$ be a Banach space and let $Y$ be a subspace of $X$. We say that $Y$ is complemented in $X$ if there exists a linear ...
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Weak* subseries convergent implies norm convergent?

Suppose $X$ is a Banach space and $X^{*}$ is separable. Suppose that $\sum x_{n}^{*}$ is a series in $X^{*}$ which has the property that every subseries $\sum x_{n_{k}}^{*}$ converges weak*. Show that ...