BANACH SPACES
Banach spaces [ edit ] Main article: Banach space Banach spaces , introduced by Stefan Banach , are complete normed vector spaces. [48] A first example is the vector space ℓ � consisting of infinite vectors with real entries � = ( � 1 , � 2 , … , � � , … ) whose � -norm ( 1 ≤ � ≤ ∞ ) given by ‖ � ‖ ∞ := sup � | � � | for � = ∞ , and ‖ � ‖ � := ( ∑ � | � � | � ) 1 � for � < ∞ . The topologies on the infinite-dimensional space ℓ � are inequivalent for different � . For example, the sequence of vectors � � = ( 2 − � , 2 − � , … , 2 − � , 0 , 0 , … ) , in which the first 2 � components are 2 − � and the following ones are 0 , converges to the zero vector for � = ∞ , but does not for � = 1 : ‖ � � ‖ ∞ = sup ( 2 − � , 0 ) = 2 − � → 0 , but ‖ � � ‖ 1 = ∑ � = 1 2 � 2 − � = 2 � ⋅ 2 − � = 1....