Projections of normed linear spaces with closed subspaces of finite codimension as kernels

Summary

It follows from [1], [4] and [7] that any closed <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"3"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"4"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"5"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"6"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"7"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"8"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"9"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"10"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"11"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"12"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"13"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"14"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>n$-codimensional subspace ($n \ge 1$ integer) of a real Banach space $X$ is the kernel of a projection $X \to X$, of norm less than $f(n) + \varepsilon$~($\varepsilon > 0$ arbitrary), where \[ f (n) = \frac{2 + (n-1) \sqrt{n+2}}{n+1}. \] We have $f(n) < \sqrt{n}$ for $n > 1$, and \[ f(n) = \sqrt{n} - \frac{1}{\sqrt{n}} + O \left(\frac{1}{n}\right). \] (The same statement, with $\sqrt{n}$ rather than $f(n)$, has been proved in [2]. A~small improvement of the statement of [2], for $n = 2$, 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 $X^*$ by adjoints of finite rank projections on $X$. 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].