Banach space
Banach algebras

A Banach algebra is a Banach space A {\displaystyle A} over K = R {\displaystyle \mathbb {K} =\mathbb {R} } or C, {\displaystyle \mathbb {C},} together with a structure of algebra over K {\displaystyle \mathbb {K} }, such that the product map A × A ∋ (a, b) ↦ a b ∈ A {\displaystyle A\times A\ni (a,b)\mapsto ab\in A} is continuous. An equivalent norm on A {\displaystyle A} can be found so that ‖ a b ‖ ≤ ‖ a ‖ ‖ b ‖ {\displaystyle \|ab\|\leq \|a\|\|b\|} for all a, b ∈ A. {\displaystyle a,b\in A.}
combination:two_responses