Questions tagged [closed-graph]

For questions about the closed graph theorem in functional analysis.

62 questions
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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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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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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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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 ...
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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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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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 ...
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 ...