zbMATH — the first resource for mathematics

Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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 in normed spaces