Explicitly, for if the right-hand side is finite, or the left-hand side is infinite. RoCookies help us deliver our services. It contains an underlying set, the subsets of this set that are feasible for measuring and the method that is used for measuring. A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. Lp spaces form an important class of Banach spaces in functional analysis, and of topological vector spaces. In both cases the embedding is continuous, in that the identity operator is a bounded linear map from Thus, we will consider ℓThe set of such functions forms a vector space, with the following natural operations:This can be made into a normed vector space in a standard way; one simply takes the The additional inner product structure allows for a richer theory, with applications to, for instance, This result may be used to prove Clarkson's inequalities, which are in turn used to establish the The situation of having no linear functionals is highly undesirable for the purposes of doing analysis. In the case of the Lebesgue measure on The vector space of (equivalence classes of) measurable functions on (This entry is from Wikipedia, the leading user-contributed encyclopedia. In many situations, the Euclidean distance is insufficient for capturing the actual distances in a given space.
One important example of a measure space is a probability space. Some later authors which is the number of non-zero entries of the vector The space of sequences has a natural vector space structure by applying addition and scalar multiplication coordinate by coordinate. The constant appearing in the above inequality is optimal, in the sense that the The situation of having no linear functionals is highly undesirable for the purposes of doing analysis. Formally, this means that the Euclidean norm of any vector is bounded by its 1-norm:For the opposite direction, the following relation between the 1-norm and the 2-norm is known:defines a subadditive function, which does define an F-norm. All rights reserved. In the case of the Lebesgue measure on The vector space of (equivalence classes of) measurable functions on Under the convention that two functions are equal if they are equal Function spaces generalizing finite-dimensional p norm spaces A measurable space consists of the first two components without a specific measure.