Lp space

These examples can be generalized; see Lp space for details.

The Lp space are one of the main objects of study of functional analysis.

Note that this does agree with our intuition since when considered as a member of an Lp space, is identified with the zero function.

On the other hand, the left-hand-side is also an inner product, this time on the Lp space .

See Lp space for details.

This can be easily generalized to characterizing the norm and spectrum of a multiplication operator on any Lp space.

Lp space, a special Banach space of functions (or rather, equivalence classes)

The wavefunction must be square-integrable on the Hilbert space (see Lp spaces) meaning:

Together with Aleksander Pełczyński, he obtained important results on the topological structure of Lp spaces.

To study this equation, consider the space that is, the Lp space of all square integrable functions in respect to the Lebesgue measure.

The proved vector space properties imply that ' is a vector subspace of Lp space.

L[a, b] (see Lp space) of square-integrable real-valued functions.

Any Lp space with the (almost everywhere) pointwise partial order is a Dedekind complete Riesz space.

Functional analysis mixes the methods of linear algebra with those of mathematical analysis and studies various function spaces, such as Lp spaces.

If (an Lp space) is the payoff of a portfolio at some future time and then we define the expected shortfall as where is the Value at risk.

Given a probability space , and letting be the Lp space in the scalar case and in d-dimensions, then we can define acceptance sets as below.

The space of all step functions on , normed by the norm (see Lp space), is a normed vector space which we denote by .

Indeed when H is considered as a distribution or an element of (see Lp space) it does not even make sense to talk of a value at zero, since such objects are only defined almost everywhere.

A set-valued risk measure is a function , where is a -dimensional Lp space, , and where is a constant solvency cone and is the set of portfolios of the reference assets.

The set of all measurable functions that are square-integrable, in the sense of the Lebesgue integral, forms a vector space which is a Hilbert space, the so-called Lp space, provided functions which are equal almost everywhere are identified.