In functional analysis, Sobczyk's theorem is a result concerning the existence of projections in Banach spaces. In its original form, the theorem states that for any separable Banach space containing the space formula_1 (of sequences converging to zero) as a subspace, there exists a projection from the ambient space onto formula_1 whose norm is at most formula_3. The theorem is not true for general non-separable Banach spaces.
A slightly modified version also commonly referred to as the Sobczyk theorem, deals with the extension of a bounded linear operator. This version asserts that if a Banach space contains a subspace that is linearly isometric to formula_1, then any bounded linear operator defined on that subspace and taking values in formula_1 can be extended to the entire space with operator norm at most twice that of the original.
The theorem is named after the American mathematician Andrew Sobczyk, who proved it in 1941.
Statement.
Original version.
The original version of the theorem states
"Let formula_6 be a separable Banach space and formula_7. Then there exists a projection formula_8 with norm at most formula_3."
Extension version.
The second version of the theorem is as follows
"Let formula_6 be a separable Banach space and let formula_11 be a subspace. If formula_12 is a bounded linear operator, then there exists an extension formula_8 with formula_14."