addition and scalar multiplication coordinate by coordinate. Explicitly, the vector sum and the scalar action for infinite sequences of real (or complex) numbers are given by: Define the -norm: Here, a complication arises, namely that the series on the right is not always convergent, so for example, the sequence made up of only ones, will have an infinite -norm for The space is then defined as the set of all infinite sequences of real (or complex) numbers such that the -norm is finite. One can check that as increases, the set grows larger. For example, the sequence is not in but it is in for as the series diverges for (the harmonic series), but is convergent for One also defines the -norm using the supremum: and the corresponding space of all bounded sequences. It turns out that if the right-hand side is finite, or the left-hand side is infinite. Thus, we will consider spaces for The -norm thus defined on is indeed a norm, and together with this norm is a Banach space. The fully general space is obtained—as seen below—by considering vectors, not only with finitely or countably-infinitely many components, but with "arbitrarily many components"; in other words, functions. An integral instead of a sum is used to define the -norm. General ℓp-space In complete analogy to the preceding definition one can define the space over a general index set (and ) as where convergence on the right means that only countably many summands are nonzero (see also Unconditional convergence). With the norm the space becomes a Banach space. In the case where is finite with elements, this construction yields with the -norm defined above. If is countably infinite, this is exactly the sequence space defined above. For uncountable sets this is a non-separable Banach space which can be seen as the locally convex direct limit of -sequence spaces. For the -norm is even induced by a canonical inner product called the , which means that holds for all vectors This inner product can expressed in terms of the norm by using the polarization identity. On it can be defined by while for the space associated with a measure space which consists of all square-integrable functions, it is Now consider the case Define where for all The index set can be turned into a measure space by giving it the discrete σ-algebra and the counting measure. Then the space