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Questions tagged [closed-graph]

For questions about the closed graph theorem in functional analysis.

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For each $x\in X$, there exists $\lim_{n \to \infty}T_n x$ and introduce $T:X\to X$ as $Tx=\lim_{n \to \infty}T_n x\quad \text{for each } x\in X$.

Let $X$ be a Banach space and let $T_n \in \mathcal{B}(X)$ for each $n \in \mathbb{N}$. Assume that for each $x \in X$, there exists $\lim_{n \to \infty} T_n x$ and introduce the mapping $T : X \to X$ ...
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Assume that $AT$ is a continuous operator. Show that $A$ is also a continuous operator.

On our lectures we have proven: Let $X$ and $Y$ be Banach spaces and let $A : X \to Y$ be a bounded, surjective operator. Show that there exists some $m > 0$ such that for every $y \in Y$ there ...
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Let $X$ be a Banach space and let $A : X \to X$ be a linear operator such that $AT - TA \in \mathcal{B}(X)$ for every $T \in \mathcal{B}(X)$.

Let $X$ be a Banach space and let $A : X \to X$ be a linear operator such that $AT - TA \in \mathcal{B}(X)$ for every $T \in \mathcal{B}(X)$. Show that $A \in \mathcal{B}(X)$. Solution. Let $0 \ne u \...
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Let $A : X \to X$ be a linear operator such that $T_f A \in \mathcal{B}(X)$ for every $f \in X^*$. Show that $A \in \mathcal{B}(X)$.

Let $X$ be a Banach space and let $0 \neq u \in X$. (a) For every $f \in X^*$, let $T_f : X \to X$ be an operator defined by the prescription $T_f x = f(x) u$. Show that $T_f \in \mathcal{B}(X)$. (b) ...
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Any $\ell^2$-closed subspace of $\ell^2 \cap \ell^1$ is finite-dimensional

Let $X$ be a closed subspace of $\ell^2$ such that $X$ is contained in $\ell^1$. It is easy to show that the inclusion operator $J \colon X \hookrightarrow \ell^1$ is closed, hence, by the closed ...
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Is a Linear Operator Bounded if its Composition with a Bounded, Injective Operator is Bounded?

You can see the question in the title and my solution is written below: Question: Let $T: X \rightarrow Y$ and $J: Y \rightarrow Z$ be linear, moreover suppose that $J$ is injective and bounded such ...
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Prove the Boundedness of a Self-Adjoint Operator in a Hilbert Space

this is the question and my attempted answer below. I would be happy if you could please proofread and let me know any suggestions if you spot any errors! :D Question: Let $H$ be a Hilbert space and $...
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Spectrum of a closbale

Consider $A: D(A) \rightarrow C_0(\mathbb{R}^3)$ where $C_0(\mathbb{R}^3)$ is the space of continuous functions vanishing at infinity and $C_C^1(\mathbb{R}^3)$ is the sub-space of $C^1$ functions with ...
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Inclusion between two complete vector subspaces continuously embedded in $L_{loc}^{1}\left(\Omega\right) $ is a continuous embedding

I have found this exercise about the closed graph theorem, but I am not yet familiar with the vector spaces of locally integrable functions. So I need your help. The aim of the exercise is to show ...
Djalal Ounadjela's user avatar
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Proving continuity of a linear map from Banach space to its dual space

The problem is Let $V$ be a Banach space over the real field, and let $T: V \to V^\ast, v \to T_v$ be a linear map between these two Banach spaces that satisfies $T_v(v)\ge 0$ for all $v\in V$. Prove ...
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When you extend a bounded, injective, and compact operator with dense image, is the extension also injective

Let $X$ and $Y$ be Banach spaces and let $V\subseteq X$ be a dense subspace. Suppose $L:V\rightarrow Y$ is a bounded, injective and compact operator with $L(V)$ dense in $Y$. Can we conclude that the ...
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Uniform Continuity is correct?

According to Heine's Theorem, "If a function is continuous on a closed and bounded interval, then the function is uniformly continuous on that interval." I understand why Heine says "...
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Closed Graph Theorem in the weak topology

Let $T$ be a bounded linear operator mapping from $E$ to itself. Assume further $T$ is closed. My question is as follows: is the graph of $T$ closed in the weak topology of $E\times E$ ? My attempt: ...
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Question about the equivalence of three versions of Closed Graph Theorem

I am studying set-valued analysis, and I saw three version of this Closed Graph Theorem. Version 1: (what I was taught in class) Let $\Gamma: X \to Y$ be a correspondence. If $\Gamma$ is closed-...
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Continuity of $H_s \subset C_0^k$ inclusion

I've been trying to show that, if $H_s \subset C_0^k$, then the inclusion is continuous. I don't have the Sobolev embedding theorem condition $s>k+n/2$. $||f||_{C_0^k} =\sum_{|\alpha|\leq k} ||\...
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Converse of the Sobolev Theorem

I am trying to solve the following problem (9.33 in Folland's book). If $H^s \subset C_0^k$, then $s>k+\frac{n}{2}$. The hint suggests to use the closed graph theorem in order to show that the ...
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Understanding the 2-norm inequality for commuting orthogonal subspaces of traceless matrices in direct sum

I am trying to investigate following problem (not only to prove it or disprove it in its current form, but also learn a "context" around it). Let $A, B$ be vector spaces of traceless ...
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Example of a closed map which is not continuous.

A map $F: X \to Y$ is said to be closed if $x_n \to x$ in $X$ and $F(x_n) \to y $ in $Y$ implies $y = F(x)$. As confusing as the definition is, I cannot seem to understand how a closed map is not ...
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How can I show that $f\mapsto f'$ is a closed map? [duplicate]

Let $X=C^1([0,1])$ and $Y=C([0,1])$ both equipped with $\|f\|=\max |f|$. Then consider $$D:X\rightarrow Y:~~f\mapsto f'$$ I want to check that $D$ is a closed map. By definition we need to show that ...
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Prove that any eigenspace of an operator T ∈ C(X ) is closed.

Definition:- An operator $T:X\to Y$ is called closed if $Γ(T)=\{(x,T(x)):x\in X\}$ is a closed set in $X ⊕Y.$ Let $C(X)$ denotes the set of all closed linear operators from a Banach space $X$ to ...
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Show $A=\{ (x,y)\in \mathbb R \mid xy=1 \}$ is closed in $\mathbb R^2.$

Claim : $A:=\{ (x,y)\in \mathbb R \mid xy=1 \}$ is closed in $\mathbb R^2.$ Hint is given, but I cannot finish. Hint Let $(a,b)\in A^c$. Then $ab\neq 1$, and WLOG : $ab>1.$ We can pick $q\in \...
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Converse of closed graph theorem

Suppose $Y$ is a normed linear space. If for every Banach space $X$ and for every linear operator $T:X\to Y$, graph of $T$ is closed implies $T$ is continuous, then can we prove that $Y$ is a Banach ...
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Is the graph of $x \mapsto \arg \min_{u \in U} f(x, u)$ closed?

Consider a continuous function $f: X \times U \to \mathbb{R}$ satisfying $f(x, \cdot)$ is convex for any $x \in X$ where $X \subset \mathbb{R}^{n}$ is compact and $U \subset \mathbb{R}^{m}$ is convex ...
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Example for closed proper convex function

I have been self-studying Proximal Algorithms by Neal Parikh and Stephen Boyd. It provides a definition of closed proper convex functions without any examples. The definition is given below. Convex ...
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How to show a correspondence is lower-hemicontinous?

I have a composite function $f:X \rightarrow Y$, such that $X\subset\mathbb{R}^n,Y\subset\mathbb{R}^{k\times k}$, where $f$ involves a combination of transformations: Form a finite number $p$ of ...
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Unbounded closed operator

Let $E,F$ be two Banach spaces and $A$ an unbouded operator from $D(A) \subset E$ to $F$. We want to prove the following lemma : Let $A$ be a closed operator. The following are equivalent : $A$ is ...
Hidda Walid's user avatar
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$\forall f\in Y^X$ with $\text{Gr}(f) $ is closed implies $f\in C(X, Y) $. Does this implies $(Y, \tau_Y) $ is compact?

$(X, \tau_X) $ and $(Y, \tau_Y) $ be two topological spaces. $\forall f\in Y^X$ with $\text{Gr}(f) $ is closed implies $f\in C(X, Y) $. Question : Does this implies $(Y, \tau_Y) $ is compact? ...
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Can we draw a closed path made up of 9 line segments , each of which intersects exactly one of the other segments?

The solution given in Fomin's book is as follows. If such a closed path were possible, then all the line segments could be partitioned into pairs of intersecting segments. But then the number of ...
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Closed graph theorem in Simon B.

Hello. I have some concern. Question 1. In this version of the closed graph theorem, the mentioned operator is defined as $T:D(T)=X\to Y$? Question 2. If I have have that an operator $T:D(T)\subset X\...
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Adjoint of an everywhere defined linear operator must be bounded?

I saw the theorem 1.8 in the book Spectral Theory of ... that If $T$ is a linear operator from the Hilbert space $H$ to another Hilbert space $H_1$, and the domain of definition of T, denoted by $\...
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For every closed surface, the set of forbidden topological minors is finite.

I want to prove that for every closed surface, the set of forbidden topological minors is finite. An idea how to do it is: by starting with the set of forbidden minors. For each forbidden minor $H$, ...
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Bounded inverse for a closed range closed operator

I am dealing with an unbounded operator $T$ on an Hilbert space $H$. I am interested in proving that it has a bounded inverse $T^{-1}$. I managed to prove that the operator is closed and self-adjoint. ...
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Is there a non-compact Hausdorff space $Y $ such that for every topological space X, every function $f:X\to Y$ with closed graph is continuous?

Closed Graph Theorem (Topological Version , necessary condition ) : $X,Y$ be two topological space where $ Y $ is a compact Hausdorff space and $f:X\to Y$ be a map with $G_f=\{(x,f(x)):x\in X\}\subset ...
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Weak convergence and continuity [check of proof]

I wanted to prove the following statement: Let $X,Y$ be Banach, $T:X\rightarrow Y$ be linear. Further for every sequence $(x_n)_{n \in \mathbb{N}}$ in X weakly convergent to $x\in X$, also $(Tx_n)_{n \...
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Closed graph theorem and convergence

Let $E,F$ be Banach spaces and let $U\subseteq F$ be a subspace and let $(T_n)$ be a sequence of bounded operators between $E$ and $F$ such that $T_n(E)\subseteq U$ for each $n\in\mathbb N$. If there ...
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Example of Continuous function into$T_1$ Space with a Graph that Isn't Closed

It is well known that the graph of a continuous function into a $T_2$ space is closed. Does this result hold if the $T_2$ condition is replaced with $T_1$? I'm guessing the answer is no, but I have ...
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Prove that an operator is bounded if and only if the operator is closable

Let $T$ be a linear densely defined operator with $T:H\rightarrow \mathbb{C^m}$ where $H$ is a Hilbert space. Prove that the operator T is bounded if and only if it is closeable. What I have tried: I ...
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About bounded and closable operators.

Let $T:H\to \mathbb{C}^m$ a densely defined operator with $H$ Hilbert space. Is it true that if $T$ is closable, then $T$ is bounded? For example the differential operator $ T:C^1[0,1]\subset C[0,1]\...
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$R(T)$ is closed iff there exists $c>0$ such that $d(x, N(T)) \leq c |T x|_F$ for all $x \in E$

I'm doing exercise 2.14.1 in Brezis' book of Functional Analysis. Could you have a check on my attempt? Let $(E, | \cdot |_E), (F, | \cdot |_F)$ be Banach spaces and $T \in \mathcal{L}(E, F)$. Let $N(...
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Is this operator bounded and closed?

Let's define the subset of $\ell^2(\mathbb C)$ $$\mathcal D(A) = \left\{ {z \in {\ell ^2}\left( C \right),\sum\limits_{k = 1}^\infty {k^2{{\left| {{z_k}} \right|}^2} < \infty } } \right\},$$ and ...
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Accumulation point of a discrete Spectrum

Let $L: \text{Dom} (L)\subset L^2(\mathbb{R}^3)\mapsto L^2(\mathbb{R}^3)$ be a self-adjoint, closed, and negative operator. I am trying to prove that the discrete spectrum $\sigma_{dis}$ of $L$ can ...
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How can I prove that this Linear operator is closed?

I know that for every infinite dimensional Banach space $X$ admits a linear and discontinuous operator. To prove it, I took a sequence $\{e_n\}_{n\geq1}$ of linearly independent vectors of $X$, then I ...
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1 vote
1 answer
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On closed-graph theorem

Let $(X, \|\cdot\|_X)$, $(Y, \|\cdot\|_Y)$ be two Banach spaces and $T: X \to Y$ a linear operator. Then: $T$ is continuous $\iff$ $T$ is closed where "$T$ closed" means its graph $G(T)\...
ric.san's user avatar
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4 votes
1 answer
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Question 2.18 from Brezis: Would someone help me to understand a solution given to this question?

I underlined parts of the solution where I'm struggling with. Would someone help me with this? In (Part 1) I did not understand the equality in (1), because from the theory, for me $$N(A^*)=\{v\in D(...
Silvinha's user avatar
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-2 votes
1 answer
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Show that graphs of a contraction map and its inverse are both closed.

The following is from Conway's book on Functional Analysis that states that being Y codomain of T Banach is necessary. Green underlined explanation can be find in here. However I am struggling the red-...
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2 votes
2 answers
311 views

Application of Closed Graph Theorem for l2 space

I have an exercise, which I am struggling with. Suppose $x\in \mathbb{R}^\mathbb{N}$ and $\forall y \in l^2$ we have $xy \in l^1$. Now I am to show that $x \in l^2$. There is a hint to use the Closed ...
analysis1's user avatar
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2 answers
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Closed graph theorem for metric spaces

I am trying to prove the following version of the closed graph theorem: Let $(X,d)$ be a complet metric space and let $K \subset M$ be compact. Let $f: K \to K$ be a function such that $\{(x,f(x)) : x ...
Luna's user avatar
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2 answers
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A linear operator $T$ is closed if and only if whenever $x_n\rightarrow x$ and $Tx_n\rightarrow y$, $y=Tx$.

Let $X,Y$ be Banach spaces, and $T:X\longrightarrow Y$ be a linear operator. I want to understand the following fact. The graph $\Gamma(T) = \{(x,Tx)\;|\;x\in X\}$ is closed in $X\oplus Y$ if and ...
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1 answer
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How to determine mathematically that an expression with one independent variable forms a closed area or not

Is there any mathematical method {other than sketching the curve from given expression (or equation)} to determine whether a given expession (or equation) forms a closed area or not? For example- ...
user215805's user avatar
1 vote
2 answers
398 views

Proof of that the graph of subdifferential is closed when $f$ is closed proper convex function on $\mathbb{R}^n$.

In the Theorem 24.4 of Convex Analysis written by R. Tyrrell Rockafellar, if $f$ is a closed proper convex function on $\mathbb{R}^n$, then the graph of $\partial f$ is a closed subset of $\mathbb{R}^...
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