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Generalized orthogonal decomposition theorem in Banach space and generalized orthogonal complemented subspace. (Chinese. English summary) Zbl 1027.46013
In a Hilbert space $H$, the Riesz orthogonal decomposition theorem says that any closed linear subspace $L\subset H$ has a unique orthogonal complement, i.e., a subspace $M$ of $H$ such that $H=L\oplus M$ and $M\perp L$. In this paper, the authors study the generalization of the Riesz theorem to Banach spaces. We call vectors $x,y$ in a normed space $X$ orthogonal and denote this by $x\perp y$, if $d_{\langle y\rangle }( x) =\|x\|$, where $\langle y\rangle $ is the linear span of $y$ and $d_{S}( x) =\inf_{s\in S}\|x-s\|$ is the shortest distance from $x$ to a set $S\subset X$, and we also call two subsets $A,B\subset X$ orthogonal and denote this by $ A\perp B$, if $d_{B}( x) =\|x\|$ for all $x\in A$. Note that in a general Banach space $X$, the orthogonal condition $A\perp B$ is not symmetric in $A,B$, but in a Hilbert space it does coincide with the traditional orthogonal condition. A (closed) linear subspace $L\subset X$ is called orthogonally complementable if there is a (closed) linear subspace $M$ with $X=L\oplus M$ and $M\perp L$. The authors obtain results on the orthogonal decomposition in Banach algebras under suitable conditions. In particular, it is shown that a closed subspace $L$ of a strictly convex space $X$ is orthogonally complementable if and only if $L$ is a Chebyshev subspace of $X$ and $F_{X}^{-1}( L^{\perp }) =\{ x\in X\mid F_{X}( x) \cap L^{\perp }\neq \emptyset \} $ is an additive set, where $F_{X}( x) =\{ x^{\ast }\in X^{\ast }\mid \langle x^{\ast },x\rangle =\|x\|^{2}\} $ and $L^{\perp }=\{ x^{\ast }\in X^{\ast }\mid \langle x^{\ast },x\rangle =0 \text{for all }x\in L\} $.

MSC:
46B20Geometry and structure of normed linear spaces
46B15Summability and bases in normed spaces