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A Note on Closedness of the Sum of Two Closed Subspaces in a Banach Space

Commun. Math. Anal.
Volume 19, Number 2 (2016), 62 - 67

A Note on Closedness of the Sum of Two Closed Subspaces in a Banach Space

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Abstract

Let $X$ be a Banach space, and $M,N$ be two closed subspaces of $X$. We collect several necessary and sufficient conditions for the closedness of $M+N$ ($M+N$ is not necessarily direct sum), and show that a necessary condition in a classical textbook is also sufficient.