CatDat

Unknown properties

There are 2 properties for which the database doesn't have an answer if they are satisfied or not. Please help to contribute the data!

• is Barr-coexact
• has effective cocongruences

Special objects

• terminal object: trivial Banach space
• initial object: trivial Banach space
• products: The product of a family of Banach spaces $(X_i,|{-}|)_{i \in I}$ is the subspace $\bigl\{x \in \prod_{i \in I} X_i : \sup_{i \in I} |x_i| < \infty\bigr\}$ of their vector space product equipped with the $\sup$-norm $|x|_\infty \coloneqq \sup_{i \in I} |x_i|$.
• coproducts: The coproduct of a family of Banach spaces $(X_i,|{-}|)_{i \in I}$ is the subspace $\bigl\{x \in \prod_{i \in I} X_i : \sum_{i \in I} |x_i| < \infty\bigr\}$ of their vector space product equipped with the $1$-norm $|x|_1 \coloneqq \sum_{i \in I} |x_i|$.

Special morphisms

• isomorphisms: bijective linear isometries
• monomorphisms: injective linear contractions
• epimorphisms: linear contractions with dense image
• regular monomorphisms: For a linear contraction $f : X \to Y$ the following are equivalent: (1) $f$ is a regular monomorphism. (2) $f$ is isometric. (3) $f$ is isomorphic to the inclusion of a closed subspace of $Y$.
• regular epimorphisms: For a linear contraction $f : X \to Y$ the following are equivalent: (1) $f$ is a regular epimorphism. (2) The norm on $Y$ is given by $|y| = \inf \{|x| : f(x) = y\}$. (3) $f$ maps the open unit ball of $X$ onto the open unit ball of $Y$.