Note on a theorem of Murray.

*(English)*Zbl 0063.03692From the introduction: In a recent paper [Trans. Am. Math. Soc. 58, 77–95 (1945; Zbl 0063.04140)] F. J. Murray has shown that in any reflexive separable Banach space \(\mathfrak B\) every closed subspace \(\mathfrak M\) admits what he calls a quasi-complement, that is, a second closed subspace \(\mathfrak N\) such that \(\mathfrak M\cap\mathfrak N = 0\) and such that \(\mathfrak M + \mathfrak N\), the smallest subspace containing both \(\mathfrak M\) and \(\mathfrak N\), is dense in \(\mathfrak B\). It is the purpose of this note to give a simpler proof of the following somewhat more general theorem.

Theorem. Let \(\mathfrak B\) be a separable normed linear space (not necessarily reflexive or even complete) and let \(\mathfrak M\) be a closed subspace of \(\mathfrak B\). Then there exists a second closed subspace \(\mathfrak N\) such that \(\mathfrak M\cap\mathfrak N = 0\) and \(\mathfrak M + \mathfrak N\) is dense in \(\mathfrak B\).

In proving this theorem it is convenient to make use of the notion of closed subspace of a linear system discussed at length in Chapter III of [S. Banach, Théorie des opérations linéaires. (1932; Zbl 0005.20901)].

Theorem. Let \(\mathfrak B\) be a separable normed linear space (not necessarily reflexive or even complete) and let \(\mathfrak M\) be a closed subspace of \(\mathfrak B\). Then there exists a second closed subspace \(\mathfrak N\) such that \(\mathfrak M\cap\mathfrak N = 0\) and \(\mathfrak M + \mathfrak N\) is dense in \(\mathfrak B\).

In proving this theorem it is convenient to make use of the notion of closed subspace of a linear system discussed at length in Chapter III of [S. Banach, Théorie des opérations linéaires. (1932; Zbl 0005.20901)].

##### MSC:

46-XX | Functional analysis |

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DOI

##### References:

[1] | F. J. Murray, Quasi-complements and closed projections in reflexive Banach spaces, Trans. Amer. Math. Soc. 58 (1945), 77 – 95. · Zbl 0063.04140 |

[2] | George W. Mackey, On infinite-dimensional linear spaces, Trans. Amer. Math. Soc. 57 (1945), 155 – 207. · Zbl 0061.24301 |

[3] | S. Banach, Théorie des operations linéaires, Warsaw, 1932. · JFM 58.0420.01 |

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