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Let p be a positive real number and let S be a Euclidean measure space equipped with Lebesgue measure. Consider the set of all measurable functions from S to C (or R) whose
absolute value to the p-th power has a finite Lebesgue integral. Identifying two such functions if they are equal almost everywhere, we obtain the set Lp(S).For f in Lp(S), we define
The space L∞(S), while related, is defined differently. We start with the set of all measurable functions from S to C (or R) which are bounded almost everywhere. By identifying two such functions if they are equal almost everywhere, we get the set L∞(S). For f in L∞(S), we set
We can also take S to be a general measure space with measure μ. Then the space Lp(S,μ), sometimes written simply Lp(μ), is the set of equivalence classes of measurable functions f from S to C such that the quantity
is finite, where two functions are equivalent if they agree μ- almost everywhere. Similarly, the space L∞(S,μ)=L∞(μ) is the set of equivalence classes of measurable functions f from S to C whose absolute values are bounded on a μ- conull set.
The most important case is when p = 2; the space L2 is a
Hilbert spaceIn mathematics, a Hilbert space is an inner product space that is complete with respect to the norm defined by the inner product. Hilbert spaces serve to clarify and generalize the concept of Fourier expansion, certain linear transformations such as the F, having major applications to Fourier seriesIn mathematics, a Fourier series named in honor of Joseph Fourier ( 1768- 1830), is a representation of a periodic function (often taken to have period 2π in a sense, the simplest case) as a sum of periodic functions of the form : which are harmonics o and quantum mechanicswavefunctions of an electron in a hydrogen atom possessing definite energy (increasing downward: n 1,2,3,. and angular momentum (increasing across: s p d . Brighter areas correspond to higher probability density for a position measurement. The angular mom, as well as other fields.If we use complex-valued functions, the space L∞ is a commutative C*-algebra with pointwise multiplication and conjugation. For many measure spaces, including all sigma-finite ones, it is in fact a commutative von Neumann algebraA von Neumann algebra is a -algebra of bounded operators on a Hilbert space which is closed in the weak operator topology, or equivalently, in the strong operator topology (under pointwise convergence) and contains the identity operator. Von Neumann algeb, since any element of L∞ defines an operatorThis article is about operators in mathematics, for other kinds of operators see operator (disambiguation). In mathematics, an operator is some kind of function; if it comes with a specified type of operand as function domain, it is no more than another w on the Hilbert space L2 by pointwise multiplication.
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