function of is defined for by If is in for some with then by Markov's inequality, A function is said to be in the space weak , or if there is a constant such that, for all The best constant for this inequality is the -norm of and is denoted by The weak coincide with the Lorentz spaces so this notation is also used to denote them. The -norm is not a true norm, since the triangle inequality fails to hold. Nevertheless, for in and in particular In fact, one has and raising to power and taking the supremum in one has Under the convention that two functions are equal if they are equal almost everywhere, then the spaces are complete . For any the expression is comparable to the -norm. Further in the case this expression defines a norm if Hence for the weak spaces are Banach spaces . A major result that uses the -spaces is the Marcinkiewicz interpolation theorem, which has broad applications to harmonic analysis and the study of singular integrals. Weighted spaces As before, consider a measure space Let be a measurable function. The -weighted space is defined as where means the measure defined by or, in terms of the Radon–Nikodym derivative, the norm for is explicitly As -spaces, the weighted spaces have nothing special, since is equal to But they are the natural framework for several results in harmonic analysis ; they appear for example in the Muckenhoupt theorem: for the classical Hilbert transform is defined on where denotes the unit circle and the Lebesgue measure; the (nonlinear) Hardy–Littlewood maximal operator is bounded on Muckenhoupt's theorem describes weights such that the Hilbert transform remains bounded on and the maximal operator on spaces on manifolds One may also define spaces on a manifold, called the intrinsic spaces of the manifold, using densities. Vector-valued spaces Given a measure space and a locally convex space (here assumed to be complete), it is possible to define spaces of -integrable -valued functions on in a number of ways. One way is to define the spaces of Bochner integrable and Pettis integrable functions, and then endow them with locally convex TVS-topologies that are (each in their own way) a natural generalization of the usual topology. Another way involves topological tensor products of with Element of the vector space are finite sums of simple tensors where each simple tensor may be