A Banach space is a complete normed vector space, a vector spaceover a subfield of the complex numberswith a normsuch that every Cauchy sequence inhas a limit in As for general vector spaces, a Banach space overis called a real Banach space, and a Banach space overis called a complex Banach space.
The familiar Euclidean spaceswith norm ofdefined byare Banach spaces. Every finite-dimensional vector space inorbecomes a Banach space on defining a norm, since all norms are equivalent on a finite-dimensionalofvector space.
The set of all continuous functionsdefined on a closed intervalbecomes a Banach space if an appropriate norm is defined in it e.g.known as the supremum norm. This is a well-defined norm since continuous functions defined on a closed interval are bounded.
Sinceis a continuous function on a closed interval, it is bounded and the supremum is attained onso the supremum is the maximum value ofon
The space is complete under this norm, and the resulting Banach space is denoted byThis example can be generalized to the spaceof all continuous functions whereis a compact space, or to the space of all bounded continuous functionswhereis any topological space, or indeed to the spaceof all bounded functionswhereis any set. In all these examples, we can multiply functions and stay in the same space: all these examples are in fact Banach algebras.
Ifis a real number, the space of all infinite sequencesof elements in such that the infinite seriesis finite. Theroot of this sum is the-norm of the sequence. The space, together with this norm, is a Banach space; it is denoted by
The Banach spaceconsists of all bounded sequences of elements inand the norm of such a sequence may be defined as the supremum of the magnitude of the sequence.
Ifare Banach spaces, then we can form their direct sumwhich has a topological vector space structure but no canonical norm, but is a Banach space for several equivalent norms, for example
This construction can be generalized to define-direct sums of arbitrarily many Banach spaces.
If is a closed linear subspace of the Banach spacethen the quotient spaceis again a Banach space.
Every inner product gives rise to an associated norm. The inner product space is called a Hilbert space if its associated norm is complete. Thus every Hilbert space is a Banach space by definition.