# 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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### 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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1 vote
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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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1 vote
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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 ...
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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. ...
1 vote
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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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### 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 ...
1 vote
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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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### 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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### 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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