Questions tagged [closed-graph]

For questions about the closed graph theorem in functional analysis.

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Showing $x \mapsto B(x,y), y\mapsto B(x,y)$ continuous $\implies B(x,y)\le c\|x\|\|y\|$ for all $x,y\in H$, where $B:H\to H$ sesquilinear, $H$ Hilbert

Let $\mathscr{H}$ be a Hilbert space and $B: \mathscr{H} \times \mathscr{H} \to \mathbb{K}$ be a sesquilinear form, and $\mathbb{K}$ denote $\mathbb{R}$ or $\mathbb{C}$. Show that 1. implies 2., ...
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23 views

Showing closeness of an operator

Let $E$ be a Banachspace, $F,G$ 2 closed, linear subspaces of $E$, such that for every $x\in E$ there are unique $y\in F, z\in G$, such that $x=y+z$. Let $T_F: E\to F, x=y+z\mapsto y$. I want to show ...
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functional analysis : problem related to closed graph theorem

enter image description here the problem above is in Conway's [Functional Analysis] (p.93) it seems to be an application of closed graph theorem if the inequality were posed the other way it could ...
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45 views

Pre-composing a Closed operator by a bounded operator

A linear map ( not necessarily bounded ) between normed linear spaces is called a closed operator if its graph is closed. Suppose $X$ is a n.l.s and $Y, Z$ are Banach spaces. Let $A : X_0 ⊆ X → Y$ be ...
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57 views

Uniform boundedness principle and closed graph Theorem

In a functional analysis class, my professor gave a problem to prove the uniform boundedness principle from the Closed graph theorem. The problem goes thus: Let $(X,\|\cdot\|_X)$ and $(Y,\|\cdot\|...
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36 views

Laplacian Operator + Closed graph theorem

I know that the Laplacian operator defined as $$\Delta:(L^2(\Omega),\|\cdot\|_{L^2(\Omega)}) \to (L^2(\Omega),\|\cdot\|_{L^2(\Omega)})$$ is unbounded. But under other settings like $$\Delta:(H^2(\...
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1answer
18 views

To show Maps are closed .Closed Graph theorem [closed]

Let $V$ and $W$ be two Banach spaces and let $T \in L(V,W)$ be surjective.Let $M$ be any subset of $V$.Prove that $T(M)$ is closed in$W$ iff $M+N(T)$ is closed in V. I tried to prove it by using ...
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1answer
21 views

About Closed Graph Theorem for conjugate linear map

Let $X$ be a Banach space over $\mathbb{C}$ and $T \colon X \to X$ be a conjugate linear map. If the graph $G(T)= \{ (x,Tx) \mid x\in X \}$ of $T$ is closed in $X\times X$, is $T$ continuous?
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29 views

Closed operator, closed graph

From a course based on Kreyszig's Introduction to Functional analysis. Let $X \neq \{0\}$ denote a complex normed vector space, and assume that the operator $T : D(T) ⊂ X → X$ is closed. Let $λ ∈ C$....
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52 views

A linear operator with a closed graph that is not bounded. [closed]

Let $D$ be a linear operator on the subspace $ S=\{(x_n):\sum_n n^2 \vert x_n \vert^2 < \infty \} \subset l_2 $ such that $ D(x_n) = (n x_n) $. How to prove that the graph of $D$ is closed but is ...
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70 views

Show that Identity is closed but not continuous

I'm trying to show the following The operator Identity defined as $$I:(C[0, 1], ||\cdot||_1) \rightarrow (C[0, 1], ||\cdot||_\infty)$$ is closed but not continuous. So $(C[0, 1], ||\cdot||_1)$ it's ...
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1answer
77 views

A problem on Closed Graph Theorem

Let $X$ and $Y$ be Banach spaces. Let $\{f_i \} \subset Y^∗$ separates points in $Y$ . Suppose that $f_i T$ is continuous for each $f_i$ , then prove that $T$ is continuous. I know that if I show ...
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122 views

Closed graph theorem seems to state that a closed operator has to be bounded?

By the closed graph theorem an operator $T$ is continuous (equivalently bounded) if and only if it its graph is closed. An operator with a closed graph is called a closed operator. So we have $$ T ...
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21 views

Regular Set of a closed Operator

Let $\mathcal{H}$ be a (separable) Hilbert space and $A : D(A) \to \mathcal{H}$ a closed densly defined operator. Define the regular set by $$\rho(A) := \{z \in \mathbb{C} : z - A \text{ bijection ...
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What is the closure of the level set of convex function $g(x):R^n\rightarrow R,$ $\{x:g(x)<a\}$.

suppose that $A:=\{x:g(x)<a\}\neq \emptyset$ and $g(x)$ is a convex function. What is the closure of $A$? My conjecture is $\bar{A}=\{x:g(x)\leq a\}$ but I do not know how to prove it..
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Why can we assume that the range of the generator of a $C_0$ semigroup is already the full space?

I am reading this script: http://www.hairer.org/notes/SPDEs.pdf about stochastic partial differential equations. I have a problem with the footnote on page 46. $L$ is the generator of some $C_0$ ...
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57 views

Show that an operator is neither closed nor closable

Let $T: D(T) \to L^p(0,1)$ for $D(T) = C[0,1] \subset L^p(0,1)$ and $p \in [1,\infty)$ be defined by $(Tf)(x) := f(0) x$ for $f \in D(T)$ and $x \in (0,1)$. Problem: We need to show that $T$ is ...
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1answer
20 views

Closed families of random variables exists?

If $\psi$ and $\nu$ are real random variables with normal distribution, their sum is also normal. Do we have such families on a half line or, more importantly on $[a,b]$? (I.e., the value of the ...
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2answers
40 views

Question regarding closeability of the operator $\Lambda:L^2(\mathbb{R})\to \mathbb{R}$ given by $\Lambda f:= f(0)$?

I have a question on the closeability of an operator. First is the definition I am using for the closeable property: Theorem Let $X$ and $Y$ be Banach spaces, let $\text{dom}(A)\subset X$ be a ...
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Rudin Functional Analysis Chapter 2 Problem 16 [duplicate]

Suppose that $X$ and $K$ are metric spaces, and $K$ is compact, and the graph of $f:X\rightarrow K$ is a closed subset of $X\times K$. Prove that $f$ is continuous. Show that compactness of $K$ cannot ...
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Closed graph theorem for operator topologies - Do operator topologies yield Fréchet spaces?

Consider the SOT and WOT operator topologies on $B(H)$, the bounded operators on a hilbert space $H$. I'm interested in the properties of the topological vector spaces induces on $B(H)$. Are they ...
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307 views

Example of a linear operator whose graph is not closed but it takes a closed set to a closed set

I want an example of a linear operator $T:X\to Y$, where $X$ and $Y$ are normed linear spaces, such that graph of $T$ is not closed but $T$ maps closed sets of $X$ to closed sets in $Y$. For ...
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203 views

Prove sublevel set is closed

$$f(x)=-\sum_{i=1}^{n}log(b_i-a_i^Tx)$$ $dom\ f_0={Ax<b}, where\ A\ is\ a\ m\ by\ n\ matrix\ with\ rows\ a_i^T$ How to prove that the sublevel sets of $f_0$ is closed? Generally, for a given ...
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1answer
162 views

Application of closed graph theorem 3

I'm trying to prove this: Let $X\;,\;Y$ Banach spaces and $ A :X\;\longrightarrow \; Y$ a linear operator. Prove using The Closed Graph Theorem , that if $GA\;\in\;X'\;\forall\;G\;\in\;Y'$, then $A\;...
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2answers
338 views

Invertible linear Operators

Consider $Y=l^1$ and $X=\{(x_n) \in Y: \sum n|x_n|<\infty\}$ and the linear operator $T:X\to Y$ by $(Tx)_n=nx_n$. I need to prove that the graph of $T$ is closed but $T$ is not continuous. The ...
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Do we really need completeness of $Y$?

Lemma: Let $T: \mathcal D(T) \to Y$ be a bounded linear operator with domain $\mathcal D(T) \subset X$, where $X$ and $Y$ are normed spaces. If $T$ is closed and $Y$ is complete, then $\mathcal D(T)$ ...
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1answer
310 views

Failure of closed graph theorem in this example

Let $(X,\|\cdot\|)$ be a separable, infinite dimensional Banach space with a Hamel basis $\{v_i,i\in I\}$ and $\|v_i\|=1$ for all $i\in I$. Define $\langle\cdot \rangle:X\to [0,\infty]$ by $\langle x\...
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1answer
83 views

Is there a function $f:X\to Y$ with closed graph such that…?

Let $X$ and $Y$ be topological spaces. Is there a function $f:X\to Y$ that is not continuous and it is not a closed map but its graph $Gr(f)$ is a closed subspace of $X\times Y$? If I think about a ...
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1answer
112 views

Show that the map $A : l^p \rightarrow l^q $ is a bounded linear map

Let $1 \leq p,q \leq \infty$ and $A= (a_{ij})$ be a scalar matrix. Suppose for every $x= x_j$, the series $\sum_{1}^{\infty}a_{ij}x_j$ is convergent for every $i$ and that $y=(y_i) \in {l}^q$ where $...
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114 views

Show continuity of a linear transformation using Closed graph theorem

Let $T$ be a linear transformation of a Banach space $B$ into a Banach Space $B'$. Let $E \subset B^{'*} $ satisfying - A) $E$ separates points of $B'$ (i.e. for for $x , y \in B' , x \neq y$ there ...
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23 views

Showing a linear map to be bounded

Let $1 \leq p,q \leq \infty$ and $A= (a_{ij})$ be a scalar matrix. Suppose for every $x= x_j$, the series $\sum_{1}^{\infty}a_{ij}x_j$ is convergent for every $i$ and that $y=(y_i) \in {l}^q$ where $...
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90 views

Showing continuity of a linear transformation using Closed graph theorem

Let $T$ be a linear transformation of a Banach space $B$ into a Banach Space $B'$. Let $E \subset B^{'*} $ satisfying - A) $E$ separates points of $B'$ (i.e. for for $x , y \in B' , x \neq y$ there ...
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1answer
759 views

The definition of a closed function and its epigraph

There is a related discussion: closed epigraphs equivalence Showing that projections $\mathbb{R}^2 \to \mathbb{R}$ are not closed My problem is rather simple: A function is closed if its epigraph ...
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427 views

Closed graph theorem; exercise

Let $E$ be a Banach space and let $T:E\to E^{\star}$ be a linear operator satisfying $\langle Tx,x\rangle\geq 0$ $\forall x\in E$. Prove that $T$ is a bounded operator. My Solution (but I have ...
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1answer
61 views

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\phi(...
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2answers
171 views

Banach-Steinhaus problem

Let be $E$ a Banach space and $(x_j)_{j=1}^{\infty}$ a sequence in $E$ such for all $\phi\in E':~~\displaystyle\sum_{j=1}^\infty|\phi(x_j)|<\infty$. Prove that $$\displaystyle\sup_{||\phi||\leq1}\...
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367 views

Closed map $T:X \to Y$ has closed graph?

Let $T:X\to Y$ be a linear operator between two normed vector spaces. My question is: If $T$ is a closed map (sends closed sets to closed), then is the graph of $T$ a closed set of $X \times Y$? ...
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1answer
29 views

Calculating the norm of $T(x)=(f_1(x),f_2(x),\dots)$

Let X be a Banach space and $f_i \in X^*, i \in \mathbb{N}$ such that $$ \sum_{i=1}^\infty|| f_i(x)|| < \infty, \forall x \in X. \hspace{2cm}(I) $$ Calculate the norm of $T: X \to l^1(\mathbb{N})...
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How to generalize this proof of the closed graph theorem

I found this tricky new proof of the closed graph theorem for a Hilbert space $H$. http://arxiv.org/pdf/1601.02600.pdf It says in the abstract, that it's possible to extend the proof to Banach space. ...
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Closed operators

I was wondering whether the following statement is true or not? If $A$ is closed, then it follows from the closed-graph theorem that it is bounded iff $D(A)$ is closed. I found this in a chapter of ...
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1answer
104 views

Closedness and continuity in infinite dimensional spaces

I cannot understand why the operator $A=d/dx: D(A)(\subset C[a,b])\to C[a,b]$ is closed when the domain $D(A)$ is chosen to be $C^1[a,b]$ while we know that we can converge to a non-differentiable ...
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126 views

Closed continuous operator has closed domain? Question about completeness

This question is about the statement: Let $X$, $Y$ be normed linear spaces and $D$ a linear subspace of $X$ and suppose that $A\colon D \to Y$ is a linear operator. If $A$ is continuous and closed ...
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1answer
436 views

Closed Graph Theorem Application

I'm having trouble working out the proof for this problem. Let $T:L^2(X)\to L^2(X)$ be a linear map such that there is another linear map $T^*:L^2(X)\to L^2(X)$ with $\langle Tu,v\rangle=\langle u,T^*...
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1answer
50 views

Show that if $B$ is compact in $X$ then $T(B)$ is closed in $Y$ and

Let $T:X\to Y$ be a closed linear operator where $X,Y$ are normed spaces.Show that if $B$ is compact in $X$ then $T(B)$ is closed in $Y$ Solution : Let $y_n=T(x_n)$ be a sequence in $T(B)$ such ...
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2answers
234 views

Unbounded Linear operator in a closed domain

I am trying to understand the Closed Graph Theorem and so I would like to see an example of an unbounded linear operator $\hat{T}: \mathscr{D}(\hat{T})\rightarrow Y$ where $\mathscr{D}(\hat{T})\subset ...
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1answer
54 views

Proof of $\hat{\mathrm{O}}$ta's theorem

I'm trying to prove $\hat{\mathrm{O}}$ta's theorem : Let $A$ be a closed operator on a Hilbert space $H$ and $\overline{\mathcal{D}(A)}=H$. Suppose that $A\mathcal{D}(A)\subset \mathcal{D}(A)$ and ...
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1answer
335 views

Closed graph implies $f$ continuous

I have a function $f \colon X \rightarrow Y$, where $X,Y$ are both compact, Hausdorff spaces, and I need to prove that if $\mathcal{G}(f)$ (the graph of $f$) is closed, then $f$ is continuous. I am ...
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1answer
448 views

Does Closed Graph Theorem imply Uniform Boundedness Principle for functionals?

For Functionals, Uniform Boundedness Principle can be rephrased as the following : Let ${X}$ be a Banach Space, $K$ be the field($\mathbb{R}$ or $\mathbb{C}$). Let $\mathcal{F}$ be the subset of $...
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1answer
459 views

About closed graph theorem

I want to show that in the closed graph theorem, the completeness of $Y$ is essential. (a.e I want to find two norm space $X,Y$ which $Y$ isn't complete and linear function $T:X\to Y$ such that $T$ is ...
2
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1answer
211 views

Closed graph theorem: How do domain and codomain affect continuity?

I had to examine the closed graph theorem under the following circumstances: $X, Y$ metric spaces with $Y$ compact. Does the theorem also hold if Y is not compact? (Assuming compactness in the ...