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Questions tagged [closed-graph]

For questions about the closed graph theorem in functional analysis.

Closed Graph Theorem: Let $X$, $Y$ be Banach spaces and $T : X \to Y$ a linear operator. Then $T$ is continuous if and only if $\Gamma = \{(x, y) \in X\times Y \mid y = T(x)\}$ is closed in $X\times Y$.

Usually, the theorem is used to show continuity of a linear operator by first demonstrating that it has a closed graph $\Gamma$.

149 questions
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When is the image of a linear operator (between Banach spaces) closed?

Let $T: X \longrightarrow Y$ be a continuous linear map between two Banach spaces. When is $\operatorname{Ran}(T)$ a closed subspace? What theorems are there? Thanks :)
Iuli
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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 $BL(X,K)$ such that for each $x \in X$, the set…
tattwamasi amrutam
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Weakening the assumptions of the Hellinger-Toeplitz theorem

The Hellinger-Toeplitz theorem states that if $T$ is a linear map from a Hilbert space $H$ to itself, satisfying $ \langle Tx,y\rangle=\langle x,Ty\rangle$ for all $ x,y \in H$, then $T$ is bounded. But can we not instead have the assumption that…
Tom
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Unbounded operator such that $P^2=P$

Does there exist an Unbounded operator $P$ on some Banach space $X$ such that $Dom(P)=X$ and $P^2=P$? If we don’t require $Dom(P)=X$, we can easily construct a Unbounded operator on $L^2[0,2π]$ by define $P$ which act on bases as…
Landau
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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 functions other than linear operators it is not that…
Anupam
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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? Notation: $Y^X$: Set of all functions from $X$ to…
SoG
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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 \ \text{bounded} \Longleftrightarrow T \…
csss
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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 aware about the Closed Graph Theorem, but I need to…
Klaramun
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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 closed but isn't bounded.) I know that when $X$…
hamid kamali
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Graph of weakly continuous linear operator

I have a few questions regarding the graph of an operator. Consider the operator $T:X \rightarrow Y$ between Banach spaces $X,Y$. Assume that $T$ is a linear operator which is (weak, weak)-continuous, so $T$ is continuous when $X$ and $Y$ are…
user103184
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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 graph theorem $J$ is bounded. Is it true that $X$ is…
Jackson Harris
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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 $$N(A^*)=\{v\in D(A^*): A^*v=0 \in E^*)\}\subset F^*$$ So,…
Silvinha
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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 linear and not bounded (by taking an example such as…
user864806
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Showing that the $T: \operatorname{dom}(T) \to \ell^{2}$ is closed

Let $T:\operatorname{dom}(T) \to \ell^{2}$ where $T(x^{n})=(mx_{m}^{n})_{m \in \mathbb N}$ Let $\operatorname{dom}(T):=\{x^{n}\in \ell^{2}: (mx_{m}^{n})_{m \in \mathbb N} \in \ell^{²}\}$ Determine whether $T$ is closed or not: Initially I attempted…
SABOY
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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 $x= (x_j)\in l^p$, the series $\sum_{1}^{\infty}a_{ij}x_j$ is convergent for every $i$ and that $y=(y_i) \in {l}^q$ where $y_i = \sum_{1}^{\infty}a_{ij}x_j =…
Dark_Knight
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