## Banach Spaces

A Banach space is a complete normed vector space, a vector space over a subfield of the complex numbers with a norm such that every Cauchy sequence in has a limit in As for general vector spaces, a Banach space over is called a real Banach space, and a Banach space over is called a complex Banach space.

The familiar Euclidean spaces with norm of defined by are Banach spaces. Every finite-dimensional vector space in or becomes a Banach space on defining a norm, since all norms are equivalent on a finite-dimensional of vector space.

The set of all continuous functions defined on a closed interval becomes 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.

Since is a continuous function on a closed interval, it is bounded and the supremum is attained on so the supremum is the maximum value of on The space is complete under this norm, and the resulting Banach space is denoted by This example can be generalized to the space of all continuous functions where is a compact space, or to the space of all bounded continuous functions where is any topological space, or indeed to the space of all bounded functions where is any set. In all these examples, we can multiply functions and stay in the same space: all these examples are in fact Banach algebras.

If is a real number, the space of all infinite sequences of elements in such that the infinite series is finite. The root 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 space consists of all bounded sequences of elements in and the norm of such a sequence may be defined as the supremum of the magnitude of the sequence.

If are Banach spaces, then we can form their direct sum which 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 space then the quotient space is 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. 