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It follows from [1], [4] and [7] that any closed \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $n$ \end{document}-codimensional subspace (\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $n \ge 1$ \end{document} integer) of a real Banach space \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $X$ \end{document} is the kernel of a projection \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $X \to X$ \end{document}, of norm less than \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $f(n) + \varepsilon$ \end{document}~(\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\varepsilon > 0$ \end{document} arbitrary), where \[ f (n) = \frac{2 + (n-1) \sqrt{n+2}}{n+1}. \] We have \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $f(n) < \sqrt{n}$ \end{document} for \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $n > 1$ \end{document}, and \[ f(n) = \sqrt{n} - \frac{1}{\sqrt{n}} + O \left(\frac{1}{n}\right). \] (The same statement, with \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\sqrt{n}$ \end{document} rather than \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $f(n)$ \end{document}, has been proved in [2]. A~small improvement of the statement of [2], for \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $n = 2$ \end{document}, is given in [3], pp.~61--62, Remark.) In [1] for this theorem a deeper statement is used, on approximations of finite rank projections on the dual space \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $X^*$ \end{document} by adjoints of finite rank projections on \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $X$ \end{document}. In this paper we show that the first cited result is an immediate consequence of the principle of local reflexivity, and of the result from [7].

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descriptions for adjoint relations, and parallel sums and differences of forms Recent Advances in Operator Theory in Hilbert and Krein spaces Oper. Theory Adv. Appl. 198 Birkhäuser Verlag Basel 211 – 227

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] Ma , M. J. and Guo , Z. M. , Homoclinic orbits for second order self-adjoint difference equations , J. Math. Anal. Appl. , 323 ( 1 ) ( 2006 ), 513 – 521 . [24

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References [1] A mberg , B. and D ickenschied , O. , On the adjoint group of a radical ring , Canad. Math. Bull. 38 no. 3 ( 1995 ), 262 – 270

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), 671 – 690 . [24] Hofmann , S ., Lu , G. Z ., Mitrea , D ., Mitrea , M . and Yan , L. X . , Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates , Mem. Amer. Math. Soc ., 214 ( 2011 ). [25

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Engineering , Springer , 2016 , 25 – 43 . [6] Anderson , D. R. , Second-order self-adjoint differential equations using a

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∈ 𝕭 . Assume that σ ( x ) is infinite and let Δ ⊂ σ ( x ) be the set of isolated points of σ ( x ). Note that Δ is nonempty. Indeed, one knows that the spectrum is totally disconnected for any self-adjoint element x in 𝕭 . Hence for every real

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titulaire de la chaire et au répétiteur qui lui serait adjoint » 46 . Mais le projet fait long feu, entre autres faute de candidat au poste de professeur qui puisse convenir aux deux partenaires. Quant à Sauvageot, Meillet 47 et Jean Marx lui signalent qu

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