Grothendieck space
Encyclopedia
In mathematics, a Grothendieck space, named for Alexander Grothendieck
, is a Banach space
X such that for all separable Banach spaces Y, every bounded linear operator from X to Y is weakly compact
, that is, the image of a bounded subset of X is a weakly compact subset of Y.
Every reflexive
Banach space is a Grothendieck space. Conversely, a separable Grothendieck space X must be reflexive, since the identity from X to X is weakly compact in this case.
Grothendieck spaces which are not reflexive include the space C(K) of all continuous functions on a Stonean
compact space K, and the space L∞(μ) for a positive measure μ (a Stonean compact space is a Hausdorff compact space in which the closure of every open set is open).
Alexander Grothendieck
Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...
, is a Banach space
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
X such that for all separable Banach spaces Y, every bounded linear operator from X to Y is weakly compact
Weak topology
In mathematics, weak topology is an alternative term for initial topology. The term is most commonly used for the initial topology of a topological vector space with respect to its continuous dual...
, that is, the image of a bounded subset of X is a weakly compact subset of Y.
Every reflexive
Reflexive space
In functional analysis, a Banach space is called reflexive if it coincides with the dual of its dual space in the topological and algebraic senses. Reflexive Banach spaces are often characterized by their geometric properties.- Normed spaces :Suppose X is a normed vector space over R or C...
Banach space is a Grothendieck space. Conversely, a separable Grothendieck space X must be reflexive, since the identity from X to X is weakly compact in this case.
Grothendieck spaces which are not reflexive include the space C(K) of all continuous functions on a Stonean
Extremally disconnected space
In mathematics, a topological space is termed extremally disconnected or extremely disconnected if the closure of every open set in it is open. An extremally disconnected space that is also compact and Hausdorff is sometimes called a Stonean space...
compact space K, and the space L∞(μ) for a positive measure μ (a Stonean compact space is a Hausdorff compact space in which the closure of every open set is open).