Questions tagged [closed-graph]
For questions about the closed graph theorem in functional analysis.
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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 ...
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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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Hellinger–Toeplitz theorem
I am trying to understand a proof of the
Hellinger–Toeplitz theorem in Kreyszig but using the Closed Graph Theorem.
Theorem. If a linear operator $T$ defined on all of the complex Hilbert space $H$ ...
user1044477
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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 ...
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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 ...
user907912
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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 ...
user1011087
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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 ...
user837101
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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)\...
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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(...
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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-...
user200918
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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 ...
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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 ...
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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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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- ...
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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 $\mathbb{R}^...
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$C^1[0,1]$ is not banach using the closed graph theory
Show that $C^1[0,1]$ is not a banach space using the closed graph theory with the maximum norm.
First, look at the derivative operator:
$D:C^1[0,1]\to C[0,1]$, $D(f)=f'$.
We can check that $D$ is ...
user864806
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A question about Banach spaces and closed graph theory
Let $X, Y$ be Banach spaces on the same field.
Let $T \in B(X,Y)$ a bijection (surjective and injective). Where $B(X,Y)$ denotes the collection of the linear, bounded operators.
Prove that: $\inf_{\|x\...
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Why is the derivative Operator defined over $C^2([-1,1])$ not closed?
The question is: Let A be the Operator defined by $A:C^2([-1,1])\subset C([-1,1])\to C(-1,1]),\; x\mapsto x'$. Show that the operator is not closed.
I suppose an example where a derivative sequence ...
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Counterexample of closed graph theorem when the image space is Hausdorff space but not compact?
Closed graph theorem on topological space requires the image space Y is compact Hausdorff. Some counter examples on Wikipedia are all non-Hausdorff which is considered more exotic examples.
What are ...
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Prove the inverse map of an injective linear function with closed graph from $U$ to $V$ (Banach spaces) is continuous iff the image is closed in $V$.
Let $U$ and $V$ be Banach spaces, let $A⊆U$ be a subspace, and let $f:A⊆U→V$ be an injective linear map with closed graph. Show that the inverse map $f^{-1} : B \to U$ is continuous if and only if $B =...
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Application of closed graph theorem: proving continuity of $F : H \to H$ satisfying $\langle{Fx, y\rangle} = \langle{x, Fy\rangle}$ [closed]
Let $H$ be a hilbert space and let $F: H \to H$ be any map. Prove by using the Closed Graph Theorem that if $\langle{Fx, y\rangle} = \langle{x, Fy\rangle}$ for all $x, y \in H$, then $F$ is continuous ...
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Closed graph theorem and continuity
Let $V,W$ be banach spaces and $T:V\to W$ be linear map.
The closed graph theorem says $T$ is continuous iff the graph of $T$ is closed .
Graph of $T$ is closed is same as saying, if ( $x_n\to x$ and $...
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