Lp space
The rooms also Lebesgue spaces in mathematics special rooms which consist of all p -fold integrable functions. The L in the name goes back to the French mathematician Henri Léon Lebesgue, as these spaces are defined using the Lebesgue integral. In the case of Banach space - valued functions ( as shown in the following for general ) refers to it as Bochner - Lebesgue spaces. The p in the label is a real parameter: For every number a - space is defined.
- 2.1 Lebesgue spaces with respect to the Lebesgue measure
- 2.2 The sequence space ℓ p
- 2.3 General ℓ p- space
- 2.4 Sobolev spaces square integrable functions
- 3.1 completeness
- 3.2 embeddings
- 3.3 Tightness and separability
- 3.4 compactness
- 3.5 Dual spaces and Reflexivity
- 4.1 Definition
- 4.2 Example
- 4.3 Extended Hilbert space
- 5.1 Definition
- 5.2 Properties
- 5.3 Example: random variable
- 6.1 Lp for p <1
- 6.2 space of locally integrable functions
- 6.3 Sobolev spaces
- 6.4 Hardy spaces
- 6.5 Lebesgue spaces on manifolds
Definition
Seminorm on ℒ p
Let be a measure space and. Then the following set is a vector space:
The through
Given figure is a semi-norm for all. The triangle inequality for this semi-norm is called Minkowski 's inequality and can be proved with the help of Hölder 's inequality.
Just then, is a norm if the empty set is the only set of measure zero in. Is there namely a zero quantity, the characteristic function is equal to the zero function, but it is true.
Factor space with norm
To get even in the case of a semi-norm on a normed space, we identify functions with each other if they are almost the same everywhere. Formally this means: People look at the ( independent ) vector subspace
And defines the space as the factor space. Two elements are therefore of equal if and only if is true, so if almost everywhere are the same.
The vector space is normalized by. The standard definition is not dependent on the representative of which is, applies to functions in the same equivalence class. This is due to the fact that the Lebesgue integral is invariant to changes in the integrand to zero quantities.
The normed vector space is complete and thus a Banach space, the norm is called the Lp- norm.
Even if one speaks of so-called - functions, it involves the entire equivalence class of a classical function. However, there are in the case of Lebesgue measure on the continuous functions two different never in the same equivalence class, so that the concept is a natural extension of the concept of continuous functions.
Special case p = ∞
Also, for you can use the essential supremum (in characters: ) define a space, the space of essentially bounded functions. There are several possibilities, but for σ - finite measure spaces all coincide. The most common is:
Is
Considering analogy to above, one obtains again a Banach space.
Examples
Lebesgue spaces with respect to the Lebesgue measure
A very important example of rooms is given by a measure space, then, is the Borel σ -algebra and the Lebesgue measure. In this connection, the shorter notation will be used.
The sequence space ℓ p
If we consider the measure space, where so than the set of natural numbers whose power set and was chosen as the counting here, then there is the space of all sequences with
For or
For.
This space is denoted by. The limiting cases and are the space of absolutely summable sequences of numbers and the space of bounded sequences. For all.
General ℓ p- space
Entirely analogously, one can consider the measure space with the counting measure for an arbitrary index set. In this case, is called the space, it is
Which may imply the convergence of the sum that only countably many summands are non-zero (see also unconditional convergence).
Sobolev spaces square integrable functions
If one chooses, as the Borel σ - algebra and, with the Lebesgue measure, then one obtains the measure space. The corresponding Lebesgue space is the space of quadratintegierbaren Sobolev functions.
Key Features
Completeness
By the theorem of Fischer- Riesz the rooms are complete for all, ie Banach spaces.
Embeddings
Is a finite measure, so true, so true of ( follows from the inequality of the generalized mean values)
For general dimensions apply for always. This is also referred to as a convex or Holder interpolation.
Tightness and separability
Be a separable measure space. Then is separable for. The room, however, is not separable in general.
Be open. For the test function space is tight in.
Compactness
The set of Kolmogorov - Riesz describes precompact or compact sets in Lp- spaces.
Dual spaces and Reflexivity
For the dual spaces of the spaces are Lebesgue spaces again. Specifically applies
Which is defined by, also is the canonical isometric isomorphism
Given by
It follows that, for the spaces are reflective.
For too isomorphic ( the isomorphism analogous to above), if σ - finite, or more generally localized. Is non- finite, then can ( again under the isomorphism ) as the Banach space of locally measurable locally essentially bounded functions represent.
The spaces and are not reflexive.
The Hilbert space L2
Definition
The room has a special role among the rooms. This can in fact provided the only with a canonical scalar product and thus becomes a Hilbert space. Be this as above, a measure space, a Hilbert space (often with the scalar product ) and
Then defined
An inner product on. The induced scalar product of this standard is the above-defined standard
Since these features of the standard are integrated according to the square, the functions are also called square integrable functions.
Example
The function which is defined by, a function of standard:
However, the function is not a function because
Other examples of functions are Schwartz functions.
Extended Hilbert space
As mentioned above, the areas are complete. So the space with the scalar product is really a Hilbert space. The space of Schwartz functions and the space of smooth functions with compact support ( a subspace of the Schwartz space ) are dense in Therefore, we obtain the inclusions
And
This is denoted by the corresponding topological dual space, especially space is called the space of tempered distributions and distributions. The pairs
Are examples of extended Hilbert spaces.
Bochner - Lebesgue spaces
The Bochner - Lebesgue spaces are a generalization of the previously considered Lebesgue spaces. They include In contrast to the Lebesgue spaces banachraumwertige functions.
Definition
Be a Banach space and a measure space. For one defines
Where " measure " refers to the Borel σ - algebra of the norm topology of. The figure
Is also a semi-norm if and only if. The Bochner - Lebesgue spaces are now the same as the Lebesgue spaces defined as the factor space.
Properties
Also the statements that are listed under properties apply to the Bochner - Lebesgue spaces. Only in the dual spaces there is a difference. For all that is
Which is defined by and the star denotes the dual space. The Bochner - Lebesgue spaces are also only be reflexive if the Banach space is reflexive.
Example: random variable
In the Stochastic considered to rooms that are equipped with a probability measure. Under a random variable is then defined as a measurable function. Further, the expected value for a quasi- integrable
Defined. Random variables, the functions are so possess a finite expectation value. Furthermore, random variables if and only in, if you can assign them a variance ( stochastic ). Since this is often required for practical applications, spaces are important in the stochastics.
The Lebesgue spaces related spaces
Often considered one also functions for addition, the Sobolev spaces and the Hardy spaces studied in functional analysis, which can be understood as special cases of the rooms and in differential geometry on manifolds there is a generalization of the rooms.
Lp for p <1
There is also a generalization of the rooms for. However, these are no more Banach spaces, because its definition does not provide a standard, but only a quasi- norm. In this case, however,
A translation invariant metric on which to make this space a complete metric vector space. The rooms are an example of a non- locally convex topological vector space.
Space of locally integrable functions
A locally integrable function is a function that does not need to be integrated in their complete domain of definition, but it must for every compact that lies in the domain to be integrated. So be open. Then a function is called locally integrable if for every compact the Lebesgue integral
Is finite. The amount of these functions is denoted by. Analogously to the spaces is formed here equivalence classes of functions, which differ by only level zero and then receives the room. With a corresponding semi-norm, this becomes a Fréchet space. This space can be understood as the space of regular distributions and can be continuously embedded in the space of distributions therefore. To let Analogously, define the spaces of locally p- integrable functions.
Sobolev spaces
Besides the already mentioned Sobolev spaces with quadratintegierbaren functions, there are other Sobolev spaces. These are defined using the weak derivatives and include p- integrable functions. These spaces are used in particular for the study of partial differential equations.
Hardy spaces
If we examine instead the measurable functions only the holomorphic or harmonic functions on integration, the corresponding spaces Hardy spaces are called.
Lebesgue spaces on manifolds
On an abstract differentiable manifold, which is not embedded in a Euclidean space, although there is no canonical extent and thus can not define functions. But still it is possible to define an analogue of the space by one instead of functions on the manifold studied so-called 1- densities. More information can be found density bundle in the article.
Swell
- Herbert Amann, Joachim Escher: Analysis. Volume 3, Birkhäuser, Basel, inter alia, 2001, ISBN 3-7643-6613-3.