Banach space (original) (raw)

Some authors use the term Banach space only in the case where X is infinite-dimensional, although on Planetmath finite-dimensional spaces are also considered to be Banach spaces.

If Y is a Banach space and X is any normed vector space, then the set of continuous linear maps f:X→Y forms a Banach space, with norm given by the operator norm. In particular, since ℝ and ℂ are complete, the continuous linear functionals on a normed vector space ℬ form a Banach space, known as the dual space of ℬ.

Examples:

• •
  Finite-dimensional normed vector spaces (http://planetmath.org/EveryFiniteDimensionalNormedVectorSpaceIsABanachSpace).
• •
  Lp spaces (http://planetmath.org/LpSpace) are by far the most common example of Banach spaces.
• •
  ℓp spaces (http://planetmath.org/Lp) are Lp spaces for the counting measure on ℕ.
• •
• •
  Finite (http://planetmath.org/FiniteMeasureSpace) signed measures on a σ-algebra (http://planetmath.org/SigmaAlgebra).