## 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:

 46B20 Geometry and structure of normed linear spaces 46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces