category of Banach spaces with linear contractions

 analysis
Notation Ban\Ban Objects Banach spaces over C\IC Morphisms linear contractions, i.e. linear maps of norm ≤1\leq 1 Related Met\Met, VectK\Vect_K External nLab link
The choice of morphisms is similar to that of Met\Met, which yields better categorical properties.

Satisfied properties

Assigned properties

Deduced properties

Unsatisfied properties

Assigned properties

Deduced properties*

*This also uses the deduced satisfied properties.

Unknown properties

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Special objects

  • terminal object: trivial Banach space
  • initial object: trivial Banach space
  • products: The product of a family of Banach spaces (Xi,∣−∣)i∈I(X_i,|{-}|)_{i \in I} is the subspace {x∈∏i∈IXi:sup⁡i∈I∣xi∣<∞}\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⁡\sup-norm ∣x∣∞≔sup⁡i∈I∣xi∣|x|_\infty \coloneqq \sup_{i \in I} |x_i|.
  • coproducts: The coproduct of a family of Banach spaces (Xi,∣−∣)i∈I(X_i,|{-}|)_{i \in I} is the subspace {x∈∏i∈IXi:∑i∈I∣xi∣<∞}\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 11-norm ∣x∣1≔∑i∈I∣xi∣|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→Yf : X \to Y the following are equivalent: (1) ff is a regular monomorphism. (2) ff is isometric. (3) ff is isomorphic to the inclusion of a closed subspace of YY.
  • regular epimorphisms: For a linear contraction f:X→Yf : X \to Y the following are equivalent: (1) ff is a regular epimorphism. (2) The norm on YY is given by ∣y∣=inf⁡{∣x∣:f(x)=y}|y| = \inf \{|x| : f(x) = y\}. (3) ff maps the open unit ball of XX onto the open unit ball of YY.