Sobolev space

Short description: Vector space of functions in mathematics

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In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of Lp-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete, i.e. a Banach space. Intuitively, a Sobolev space is a space of functions possessing sufficiently many derivatives for some application domain, such as partial differential equations, and equipped with a norm that measures both the size and regularity of a function.

Sobolev spaces are named after the Russian mathematician Sergei Sobolev. Their importance comes from the fact that weak solutions of some important partial differential equations exist in appropriate Sobolev spaces, even when there are no strong solutions in spaces of continuous functions with the derivatives understood in the classical sense.

## Contents

* 1 Motivation
* 2 Sobolev spaces with integer k
* 2.1 One-dimensional case
* 2.1.1 The case p = 2
* 2.1.2 Other examples
* 2.2 Multidimensional case
* 2.2.1 Approximation by smooth functions
* 2.2.2 Examples
* 2.2.3 Absolutely continuous on lines (ACL) characterization of Sobolev functions
* 2.2.4 Functions vanishing at the boundary
* 3 Traces
* 4 Sobolev spaces with non-integer k
* 4.1 Bessel potential spaces
* 4.2 Sobolev–Slobodeckij spaces
* 5 Extension operators
* 5.1 Case of p = 2
* 5.2 Extension by zero
* 6 Sobolev embeddings
* 7 See also
* 8 Notes
* 9 References
* 10 External links

## Motivation

In this section and throughout the article Ω is an open subset of ℝn.

There are many criteria for smoothness of mathematical functions. The most basic criterion may be that of continuity. A stronger notion of smoothness is that of differentiability (because functions that are differentiable are also continuous) and a yet stronger notion of smoothness is that the derivative also be continuous (these functions are said to be of class C1 -- see Differentiability classes). Differentiable functions are important in many areas, and in particular for differential equations. In the twentieth century, however, it was observed that the space C1 (or C2, etc.) was not exactly the right space to study solutions of differential equations. The Sobolev spaces are the modern replacement for these spaces in which to look for solutions of partial differential equations.

Quantities or properties of the underlying model of the differential equation are usually expressed in terms of integral norms. A typical example is measuring the energy of a temperature or velocity distribution by an L2-norm. It is therefore important to develop a tool for differentiating Lebesgue space functions.

The integration by parts formula yields that for every u∈Ck(Ω), where k is a natural number, and for all infinitely differentiable functions with compact support φ∈Cc∞(Ω),

∫ΩuDαφdx=(−1)|α|∫ΩφDαudx,

where α=(α1,...,αn) is a multi-index of order |α|=k and we are using the notation:

Dαf=∂|α|f∂x1α1…∂xnαn.

The left-hand side of this equation still makes sense if we assume u to be only locally integrable. If there exists a locally integrable function v, such that

∫ΩuDαφdx=(−1)|α|∫Ωφvdxfor all φ∈Cc∞(Ω),

then we call v the weak α-th partial derivative of u. If there exists a weak α-th partial derivative of u, then it is uniquely defined almost everywhere, and thus it is uniquely determined as an element of a Lebesgue space. On the other hand, if u∈Ck(Ω), then the classical and the weak derivative coincide. Thus, if v is a weak α-th partial derivative of u, we may denote it by Dαu:=v.

For example, the function

u(x)={1+x−1<x<010x=01−x0<x<10else

is not continuous at zero, and not differentiable at −1, 0, or 1. Yet the function

v(x)={1−1<x<0−10<x<10else

satisfies the definition for being the weak derivative of u(x), which then qualifies as being in the Sobolev space W1,p (for any allowed p, see definition below).

The Sobolev spaces Wk,p(Ω) combine the concepts of weak differentiability and Lebesgue norms.

## Sobolev spaces with integer k

### One-dimensional case

In the one-dimensional case the Sobolev space Wk,p(ℝ) for 1≤p≤∞ is defined as the subset of functions f in Lp(ℝ) such that f and its weak derivatives up to order k have a finite Lp norm. As mentioned above, some care must be taken to define derivatives in the proper sense. In the one-dimensional problem it is enough to assume that the (k−1)-th derivative f(k−1) is differentiable almost everywhere and is equal almost everywhere to the Lebesgue integral of its derivative (this excludes irrelevant examples such as Cantor's function).

With this definition, the Sobolev spaces admit a natural norm,

‖f‖k,p=(∑i=0k‖f(i)‖pp)1p=(∑i=0k∫|f(i)(t)|pdt)1p.

One can extend this to the case p=∞, with the norm then defined using the essential supremum by

‖f‖k,∞=maxi=0,…,k‖f(i)‖∞=maxi=0,…,k(esssupt|f(i)(t)|).

Equipped with the norm ‖⋅‖k,p,Wk,p becomes a Banach space. It turns out that it is enough to take only the first and last in the sequence, i.e., the norm defined by

‖f(k)‖p+‖f‖p

is equivalent to the norm above (i.e., the induced topologies of the norms are the same).

#### The case p = 2

Sobolev spaces with p = 2 are especially important because of their connection with Fourier series and because they form a Hilbert space. A special notation has arisen to cover this case, since the space is a Hilbert space:

Hk=Wk,2.

The space Hk can be defined naturally in terms of Fourier series whose coefficients decay sufficiently rapidly, namely,

Hk(𝕋)={f∈L2(𝕋):∑n=−∞∞(1+n2+n4+…+n2k)|f^(n)|2<∞},

where f^ is the Fourier series of f, and 𝕋 denotes the 1-torus. As above, one can use the equivalent norm

‖f‖k,22=∑n=−∞∞(1+|n|2)k|f^(n)|2.

Both representations follow easily from Parseval's theorem and the fact that differentiation is equivalent to multiplying the Fourier coefficient by in.

Furthermore, the space Hk admits an inner product, like the space H0=L2. In fact, the Hk inner product is defined in terms of the L2 inner product:

⟨u,v⟩Hk=∑i=0k⟨Diu,Div⟩L2.

The space Hk becomes a Hilbert space with this inner product.

#### Other examples

In one dimension, some other Sobolev spaces permit a simpler description. For example, W1,1(0,1) is the space of absolutely continuous functions on (0, 1) (or rather, equivalence classes of functions that are equal almost everywhere to such), while W1,∞(I) is the space of bounded Lipschitz functions on I, for every interval I. However, these properties are lost or not as simple for functions of more than one variable.

All spaces Wk,∞ are (normed) algebras, i.e. the product of two elements is once again a function of this Sobolev space, which is not the case for p<∞. (E.g., functions behaving like |x|−1/3 at the origin are in L2, but the product of two such functions is not in L2).

### Multidimensional case

The transition to multiple dimensions brings more difficulties, starting from the very definition. The requirement that f(k−1) be the integral of f(k) does not generalize, and the simplest solution is to consider derivatives in the sense of distribution theory.

A formal definition now follows. Let k∈ℕ,1⩽p⩽∞. The Sobolev space Wk,p(Ω) is defined to be the set of all functions f on Ω such that for every multi-index α with |α|⩽k, the mixed partial derivative

f(α)=∂|α|f∂x1α1…∂xnαn

exists in the weak sense and is in Lp(Ω), i.e.

‖f(α)‖Lp<∞.

That is, the Sobolev space Wk,p(Ω) is defined as

Wk,p(Ω)={u∈Lp(Ω):Dαu∈Lp(Ω)∀|α|⩽k}.

The natural number k is called the order of the Sobolev space Wk,p(Ω).

There are several choices for a norm for Wk,p(Ω). The following two are common and are equivalent in the sense of equivalence of norms:

‖u‖Wk,p(Ω):={(∑|α|⩽k‖Dαu‖Lp(Ω)p)1p1⩽p<∞;max|α|⩽k‖Dαu‖L∞(Ω)p=∞;

and

‖u‖'Wk,p(Ω):={∑|α|⩽k‖Dαu‖Lp(Ω)1⩽p<∞;∑|α|⩽k‖Dαu‖L∞(Ω)p=∞.

With respect to either of these norms, Wk,p(Ω) is a Banach space. For p<∞,Wk,p(Ω) is also a separable space. It is conventional to denote Wk,2(Ω) by Hk(Ω) for it is a Hilbert space with the norm ‖⋅‖Wk,2(Ω).[1]

#### Approximation by smooth functions

It is rather hard to work with Sobolev spaces relying only on their definition. It is therefore interesting to know that by the Meyers–Serrin theorem a function u∈Wk,p(Ω) can be approximated by smooth functions. This fact often allows us to translate properties of smooth functions to Sobolev functions. If p is finite and Ω is open, then there exists for any u∈Wk,p(Ω) an approximating sequence of functions um∈C∞(Ω) such that:

‖um−u‖Wk,p(Ω)->0.

If Ω has Lipschitz boundary, we may even assume that the um are the restriction of smooth functions with compact support on all of ℝn.[2]

#### Examples

In higher dimensions, it is no longer true that, for example, W1,1 contains only continuous functions. For example, |x|−1∈W1,1(𝔹3) where 𝔹3 is the unit ball in three dimensions. For k>n/p, the space Wk,p(Ω) will contain only continuous functions, but for which k this is already true depends both on p and on the dimension. For example, as can be easily checked using spherical polar coordinates for the function f:𝔹n->ℝ∪{∞} defined on the n-dimensional unit ball we have:

f(x)=|x|−α∈Wk,p(𝔹n)⟺α<np−k.

Intuitively, the blow-up of f at 0 "counts for less" when n is large since the unit ball has "more outside and less inside" in higher dimensions.

#### Absolutely continuous on lines (ACL) characterization of Sobolev functions

Let 1⩽p⩽∞. If a function is in W1,p(Ω), then, possibly after modifying the function on a set of measure zero, the restriction to almost every line parallel to the coordinate directions in ℝn is absolutely continuous; what's more, the classical derivative along the lines that are parallel to the coordinate directions are in Lp(Ω). Conversely, if the restriction of f to almost every line parallel to the coordinate directions is absolutely continuous, then the pointwise gradient ∇f exists almost everywhere, and f is in W1,p(Ω) provided f,|∇f|∈Lp(Ω). In particular, in this case the weak partial derivatives of f and pointwise partial derivatives of f agree almost everywhere. The ACL characterization of the Sobolev spaces was established by Otto M. Nikodym (1933); see (Maz'ya 2011).

A stronger result holds when p>n. A function in W1,p(Ω) is, after modifying on a set of measure zero, Hölder continuous of exponent γ=1−np, by Morrey's inequality. In particular, if p=∞ and Ω has Lipschitz boundary, then the function is Lipschitz continuous.

#### Functions vanishing at the boundary

See also: Trace operator

The Sobolev space W1,2(Ω) is also denoted by H1(Ω). It is a Hilbert space, with an important subspace H01(Ω) defined to be the closure of the infinitely differentiable functions compactly supported in Ω in H1(Ω). The Sobolev norm defined above reduces here to

‖f‖H1=(∫Ω|f|2+|∇f|2)12.

When Ω has a regular boundary, H01(Ω) can be described as the space of functions in H1(Ω) that vanish at the boundary, in the sense of traces (see below). When n=1, if Ω=(a,b) is a bounded interval, then H01(a,b) consists of continuous functions on [a,b] of the form

f(x)=∫axf′(t)dt,x∈[a,b]

where the generalized derivative f′ is in L2(a,b) and has 0 integral, so that f(b)=f(a)=0.

When Ω is bounded, the Poincaré inequality states that there is a constant C=C(Ω) such that:

∫Ω|f|2⩽C2∫Ω|∇f|2,f∈H01(Ω).

When Ω is bounded, the injection from H01(Ω) to L2(Ω), is compact. This fact plays a role in the study of the Dirichlet problem, and in the fact that there exists an orthonormal basis of L2(Ω) consisting of eigenvectors of the Laplace operator (with Dirichlet boundary condition).

## Traces

See also: Trace operator

Sobolev spaces are often considered when investigating partial differential equations. It is essential to consider boundary values of Sobolev functions. If u∈C(Ω), those boundary values are described by the restriction u|∂Ω. However, it is not clear how to describe values at the boundary for u∈Wk,p(Ω), as the n-dimensional measure of the boundary is zero. The following theorem[2] resolves the problem:

Trace theorem -- Assume Ω is bounded with Lipschitz boundary. Then there exists a bounded linear operator T:W1,p(Ω)->Lp(∂Ω) such that Tu=u|∂Ωu∈W1,p(Ω)∩C(Ω‾)‖Tu‖Lp(∂Ω)⩽c(p,Ω)‖u‖W1,p(Ω)u∈W1,p(Ω).

Tu is called the trace of u. Roughly speaking, this theorem extends the restriction operator to the Sobolev space W1,p(Ω) for well-behaved Ω. Note that the trace operator T is in general not surjective, but for 1 < p < ∞ it maps continuously onto the Sobolev–Slobodeckij space W1−1p,p(∂Ω).

Intuitively, taking the trace costs 1/p of a derivative. The functions u in W1,p(Ω) with zero trace, i.e. Tu = 0, can be characterized by the equality

W01,p(Ω)={u∈W1,p(Ω):Tu=0},

where

W01,p(Ω):={u∈W1,p(Ω):∃{um}m=1∞⊂Cc∞(Ω), such that um->u in W1,p(Ω)}.

In other words, for Ω bounded with Lipschitz boundary, trace-zero functions in W1,p(Ω) can be approximated by smooth functions with compact support.

## Sobolev spaces with non-integer k

### Bessel potential spaces

For a natural number k and 1 < p < ∞ one can show (by using Fourier multipliers[3][4]) that the space Wk,p(ℝn) can equivalently be defined as

Wk,p(ℝn)=Hk,p(ℝn):={f∈Lp(ℝn):ℱ−1[(1+|ξ|2)k2ℱf]∈Lp(ℝn)},

with the norm

‖f‖Hk,p(ℝn):=‖ℱ−1[(1+|ξ|2)k2ℱf]‖Lp(ℝn).

This motivates Sobolev spaces with non-integer order since in the above definition we can replace k by any real number s. The resulting spaces

Hs,p(ℝn):={f∈𝒮′(ℝn):ℱ−1[(1+|ξ|2)s2ℱf]∈Lp(ℝn)}

are called Bessel potential spaces[5] (named after Friedrich Bessel). They are Banach spaces in general and Hilbert spaces in the special case p = 2.

For s≥0,Hs,p(Ω) is the set of restrictions of functions from Hs,p(ℝn) to Ω equipped with the norm

‖f‖Hs,p(Ω):=inf⁡{‖g‖Hs,p(ℝn):g∈Hs,p(ℝn),g|Ω=f}.

Again, Hs,p(Ω) is a Banach space and in the case p = 2 a Hilbert space.

Using extension theorems for Sobolev spaces, it can be shown that also Wk,p(Ω) = Hk,p(Ω) holds in the sense of equivalent norms, if Ω is domain with uniform Ck-boundary, k a natural number and 1 < p < ∞. By the embeddings

Hk+1,p(ℝn)↪Hs′,p(ℝn)↪Hs,p(ℝn)↪Hk,p(ℝn),k⩽s⩽s′⩽k+1

the Bessel potential spaces Hs,p(ℝn) form a continuous scale between the Sobolev spaces Wk,p(ℝn). From an abstract point of view, the Bessel potential spaces occur as complex interpolation spaces of Sobolev spaces, i.e. in the sense of equivalent norms it holds that

[Wk,p(ℝn),Wk+1,p(ℝn)]θ=Hs,p(ℝn),

where:

1⩽p⩽∞, 0<θ<1, s=(1−θ)k+θ(k+1)=k+θ.

### Sobolev–Slobodeckij spaces

Another approach to define fractional order Sobolev spaces arises from the idea to generalize the Hölder condition to the Lp-setting.[6] For 1⩽p<∞,θ∈(0,1) and f∈Lp(Ω), the Slobodeckij seminorm (roughly analogous to the Hölder seminorm) is defined by

[f]θ,p,Ω:=(∫Ω∫Ω|f(x)−f(y)|p|x−y|θp+ndxdy)1p.

Let s > 0 be not an integer and set θ=s−⌊s⌋∈(0,1). Using the same idea as for the Hölder spaces, the Sobolev–Slobodeckij space[7] Ws,p(Ω) is defined as

Ws,p(Ω):={f∈W⌊s⌋,p(Ω):sup|α|=⌊s⌋[Dαf]θ,p,Ω<∞}.

It is a Banach space for the norm

‖f‖Ws,p(Ω):=‖f‖W⌊s⌋,p(Ω)+sup|α|=⌊s⌋[Dαf]θ,p,Ω.

If Ω is suitably regular in the sense that there exist certain extension operators, then also the Sobolev–Slobodeckij spaces form a scale of Banach spaces, i.e. one has the continuous injections or embeddings

Wk+1,p(Ω)↪Ws′,p(Ω)↪Ws,p(Ω)↪Wk,p(Ω),k⩽s⩽s′⩽k+1.

There are examples of irregular Ω such that W1,p(Ω) is not even a vector subspace of Ws,p(Ω) for 0 < s < 1 (see Example 9.1 of [8])

From an abstract point of view, the spaces Ws,p(Ω) coincide with the real interpolation spaces of Sobolev spaces, i.e. in the sense of equivalent norms the following holds:

Ws,p(Ω)=(Wk,p(Ω),Wk+1,p(Ω))θ,p,k∈ℕ,s∈(k,k+1),θ=s−⌊s⌋.

Sobolev–Slobodeckij spaces play an important role in the study of traces of Sobolev functions. They are special cases of Besov spaces.[4]

The constant arising in the characterization of the fractional Sobolev space Ws,p(Ω) can be characterized through the Bourgain-Brezis-Mironescu formula:

lims↗1(1−s)∫Ω∫Ω|f(x)−f(y)|p|x−y|sp+ndxdy=2πn−12Γ(p+12)pΓ(p+n2)∫Ω|∇f|p;

and the condition

lim sups↗1(1−s)∫Ω∫Ω|f(x)−f(y)|p|x−y|sp+ndxdy<∞

characterizes those functions of Lp(Ω) that are in the first-order Sobolev space W1,p(Ω).[9]

## Extension operators

If Ω is a domain whose boundary is not too poorly behaved (e.g., if its boundary is a manifold, or satisfies the more permissive "cone condition") then there is an operator A mapping functions of Ω to functions of ℝn such that:

1. Au(x) = u(x) for almost every x in Ω and
2. A:Wk,p(Ω)->Wk,p(ℝn) is continuous for any 1 ≤ p ≤ ∞ and integer k.

We will call such an operator A an extension operator for Ω.

### Case of p = 2

Extension operators are the most natural way to define Hs(Ω) for non-integer s (we cannot work directly on Ω since taking Fourier transform is a global operation). We define Hs(Ω) by saying that u∈Hs(Ω) if and only if Au∈Hs(ℝn). Equivalently, complex interpolation yields the same Hs(Ω) spaces so long as Ω has an extension operator. If Ω does not have an extension operator, complex interpolation is the only way to obtain the Hs(Ω) spaces.

As a result, the interpolation inequality still holds.

### Extension by zero

Like above, we define H0s(Ω) to be the closure in Hs(Ω) of the space Cc∞(Ω) of infinitely differentiable compactly supported functions. Given the definition of a trace, above, we may state the following

Theorem -- Let Ω be uniformly Cm regular, m ≥ s and let P be the linear map sending u in Hs(Ω) to (u,dudn,…,dkudnk)|G where d/dn is the derivative normal to G, and k is the largest integer less than s. Then H0s is precisely the kernel of P.

If u∈H0s(Ω) we may define its extension by zero u~∈L2(ℝn) in the natural way, namely

u~(x)={u(x)x∈Ω0else

Theorem -- Let s>12. The map u↦u~ is continuous into Hs(ℝn) if and only if s is not of the form n+12 for n an integer.

For f ∈ Lp(Ω) its extension by zero,

Ef:={fon Ω,0otherwise

is an element of Lp(ℝn). Furthermore,

‖Ef‖Lp(ℝn)=‖f‖Lp(Ω).

In the case of the Sobolev space W1,p(Ω) for 1 ≤ p ≤ ∞, extending a function u by zero will not necessarily yield an element of W1,p(ℝn). But if Ω is bounded with Lipschitz boundary (e.g. ∂Ω is C1), then for any bounded open set O such that Ω⊂⊂O (i.e. Ω is compactly contained in O), there exists a bounded linear operator[2]

E:W1,p(Ω)->W1,p(ℝn),

such that for each u∈W1,p(Ω):Eu=u a.e. on Ω, Eu has compact support within O, and there exists a constant C depending only on p, Ω, O and the dimension n, such that

‖Eu‖W1,p(ℝn)⩽C‖u‖W1,p(Ω).

We call Eu an extension of u to ℝn.

## Sobolev embeddings

Main article: Sobolev inequality

It is a natural question to ask if a Sobolev function is continuous or even continuously differentiable. Roughly speaking, sufficiently many weak derivatives (i.e. large k) result in a classical derivative. This idea is generalized and made precise in the Sobolev embedding theorem.

Write Wk,p for the Sobolev space of some compact Riemannian manifold of dimension n. Here k can be any real number, and 1 ≤ p ≤ ∞. (For p = ∞ the Sobolev space Wk,∞ is defined to be the Hölder space Cn,α where k = n + α and 0 < α ≤ 1.) The Sobolev embedding theorem states that if k⩾m and k−np⩾m−nq then

Wk,p⊆Wm,q

and the embedding is continuous. Moreover, if k>m and k−np>m−nq then the embedding is completely continuous (this is sometimes called Kondrachov's theorem or the Rellich–Kondrachov theorem). Functions in Wm,∞ have all derivatives of order less than m continuous, so in particular this gives conditions on Sobolev spaces for various derivatives to be continuous. Informally these embeddings say that to convert an Lp estimate to a boundedness estimate costs 1/p derivatives per dimension.

There are similar variations of the embedding theorem for non-compact manifolds such as ℝn (Stein 1970). Sobolev embeddings on ℝn that are not compact often have a related, but weaker, property of cocompactness.

## See also

* Sobolev mapping
* Souček space
* Besov space
* Triebel–Lizorkin space

## Notes

1. ↑ Evans 2010, Chapter 5.2
2. ↑ 2.0 2.1 2.2 Adams & Fournier 2003
3. ↑ Bergh & Löfström 1976
4. ↑ 4.0 4.1 Triebel 1995
5. ↑ Bessel potential spaces with variable integrability have been independently introduced by Almeida & Samko (A. Almeida and S. Samko, "Characterization of Riesz and Bessel potentials on variable Lebesgue spaces", J. Function Spaces Appl. 4 (2006), no. 2, 113–144) and Gurka, Harjulehto & Nekvinda (P. Gurka, P. Harjulehto and A. Nekvinda: "Bessel potential spaces with variable exponent", Math. Inequal. Appl. 10 (2007), no. 3, 661–676).
6. ↑ Lunardi 1995
7. ↑ In the literature, fractional Sobolev-type spaces are also called Aronszajn spaces, Gagliardo spaces or Slobodeckij spaces, after the names of the mathematicians who introduced them in the 1950s: N. Aronszajn ("Boundary values of functions with finite Dirichlet integral", Techn. Report of Univ. of Kansas 14 (1955), 77–94), E. Gagliardo ("Proprietà di alcune classi di funzioni in più variabili", Ricerche Mat. 7 (1958), 102–137), and L. N. Slobodeckij ("Generalized Sobolev spaces and their applications to boundary value problems of partial differential equations", Leningrad. Gos. Ped. Inst. Učep. Zap. 197 (1958), 54–112).
8. ↑ Di Nezza, Eleonora; Palatucci, Giampiero; Valdinoci, Enrico (2012-07-01). "Hitchhikerʼs guide to the fractional Sobolev spaces" (in en). Bulletin des Sciences Mathématiques 136 (5): 521–573. doi:10.1016/j.bulsci.2011.12.004. ISSN 0007-4497.
9. ↑ Bourgain, Jean; Brezis, Haïm; Mironescu, Petru (2001). "Another look at Sobolev spaces". in Menaldi, José Luis. Optimal control and partial differential equations. In honour of Professor Alain Bensoussan's 60th birthday. Proceedings of the conference, Paris, France, December 4, 2000. Amsterdam: IOS Press; Tokyo: Ohmsha. pp. 439–455. ISBN 978-1-58603-096-4.

## References

* Adams, Robert A.; Fournier, John (2003). Sobolev Spaces. Pure and Applied Mathematics. 140 (2nd ed.). Boston, MA: Academic Press. ISBN 978-0-12-044143-3. .
* Aubin, Thierry (1982), Nonlinear analysis on manifolds. Monge-Ampère equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 252, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-4612-5734-9, ISBN 978-0-387-90704-8 .
* Bergh, Jöran; Löfström, Jörgen (1976), Interpolation Spaces, An Introduction, Grundlehren der Mathematischen Wissenschaften, 223, Springer-Verlag, pp. X + 207, ISBN 978-7-5062-6011-4
* Evans, Lawrence C. (2010). Partial Differential Equations. Graduate Studies in Mathematics. 19 (2nd ed.). American Mathematical Society. pp. 749. ISBN 978-0-8218-4974-3.
* Leoni, Giovanni (2009). A First Course in Sobolev Spaces. Graduate Studies in Mathematics. 105. American Mathematical Society. pp. xvi+607. ISBN 978-0-8218-4768-8.
* Maz'ja, Vladimir G. (1985), Sobolev Spaces, Springer Series in Soviet Mathematics, Berlin–Heidelberg–New York: Springer-Verlag, pp. xix+486, doi:10.1007/978-3-662-09922-3, ISBN 0-387-13589-8
* Maz'ya, Vladimir G.; Poborchi, Sergei V. (1997), Differentiable Functions on Bad Domains, Singapore–New Jersey–London–Hong Kong: World Scientific, pp. xx+481, ISBN 981-02-2767-1, https://books.google.com/books?id=ZeGwVyRtza4C .
* Maz'ya, Vladimir G. (2011), Sobolev Spaces. With Applications to Elliptic Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften, 342 (2nd revised and augmented ed.), Berlin–Heidelberg–New York: Springer Verlag, pp. xxviii+866, doi:10.1007/978-3-642-15564-2, ISBN 978-3-642-15563-5, https://books.google.com/books?id=vNx2I5xu1AAC .
* Lunardi, Alessandra (1995), Analytic semigroups and optimal regularity in parabolic problems, Basel: Birkhäuser Verlag .
* Nikodym, Otto (1933), "Sur une classe de fonctions considérée dans l'étude du problème de Dirichlet", Fund. Math. 21: 129–150, doi:10.4064/fm-21-1-129-150, http://minidml.mathdoc.fr/cgi-bin/location?id=00113509 .
* Hazewinkel, Michiel, ed. (2001), "Imbedding theorems", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Imbedding_theorems&oldid=14600 .
* Hazewinkel, Michiel, ed. (2001), "Sobolev space", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Sobolev_space&oldid=17396 .
* Sobolev, S. L. (1963), "On a theorem of functional analysis", Eleven Papers on Analysis, American Mathematical Society Translations: Series 2, 34, pp. 39–68, doi:10.1090/trans2/034/02, ISBN 9780821817346 ; translation of Mat. Sb., 4 (1938) pp. 471–497.
* Sobolev, S.L. (1963), Some applications of functional analysis in mathematical physics, Amer. Math. Soc. .
* Stein, E (1970), Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, ISBN 0-691-08079-8, https://archive.org/details/singularintegral0000stei .
* Triebel, H. (1995), Interpolation Theory, Function Spaces, Differential Operators, Heidelberg: Johann Ambrosius Barth .
* Ziemer, William P. (1989), Weakly differentiable functions, Graduate Texts in Mathematics, 120, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-4612-1015-3, ISBN 978-0-387-97017-2 .

## External links

* Eleonora Di Nezza, Giampiero Palatucci, Enrico Valdinoci (2011). "Hitchhiker's guide to the fractional Sobolev spaces".

* v
* t
* e

Functional analysis (topics)

Topological vector spaces|

* Asplund
* Banach (list)
* Banach lattice
* Barrelled
* Bornological
* Brauner
* F-space
* Fréchet (tame)
* Hilbert (Inner product space
* Polarization identity)
* LF-space
* Locally convex (Seminorms/Minkowski functionals)
* Mackey
* Montel
* Nuclear
* Normed (norm)
* Quasinormed
* Reflexive
* Riesz
* Smith
* Stereotype
* Strictly convex
* Webbed
* Topological tensor product (of Hilbert spaces)

Topologies of function spaces|

* Dual
* Dual space (Dual norm)
* Operator
* Ultraweak
* Weak (polar
* operator)
* Mackey
* Strong (polar
* operator)
* Ultrastrong
* Uniform convergence

Linear operators|

* Adjoint
* Bilinear (form
* operator
* sesquilinear)
* (Un)Bounded
* Closed
* Compact (on Hilbert spaces)
* (Dis)Continuous
* Densely defined
* Fredholm
* Hilbert–Schmidt
* Functionals (positive)
* Normal
* Nuclear
* Self-adjoint
* Strictly singular
* Trace class
* Transpose
* Unitary

Operator theory|

* Banach algebras
* C*-algebras
* Spectrum (C*-algebra
* radius)
* Spectral theory (of ODEs
* Spectral theorem)
* Polar decomposition
* Singular value decomposition

Theorems|

* Banach–Alaoglu
* Banach–Mazur
* Banach–Saks
* Banach–Schauder (open mapping)
* Banach–Steinhaus (Uniform boundedness)
* Bessel's inequality
* Cauchy–Schwarz inequality
* Closed graph
* Closed range
* Eberlein–Šmulian
* Freudenthal spectral
* Gelfand–Mazur
* Gelfand–Naimark
* Goldstine
* Hahn–Banach (hyperplane separation)
* Kakutani fixed-point
* Krein–Milman
* Lomonosov's invariant subspace
* Mackey–Arens
* Mazur's lemma
* M. Riesz extension
* Riesz representation
* Parseval's identity
* Schauder fixed-point

Analysis|

* Abstract Wiener space
* Bochner space
* Differentiation in Fréchet spaces
* Derivatives (Fréchet
* Gateaux
* functional
* holomorphic)
* Integrals (Bochner
* Dunford
* Gelfand–Pettis
* regulated
* Paley–Wiener
* weak)
* Functional calculus (Borel
* continuous
* holomorphic)
* Inverse function theorem (Nash–Moser theorem)
* Measures (Lebesgue
* Projection-valued
* Vector)
* Weakly measurable function

Types of sets|

* Absolutely convex
* Absorbing
* Balanced
* Bounded
* Convex
* Convex cone (subset)
* Linear cone (subset)
* Radial
* Star-shaped
* Symmetric
* Zonotope

Subsets / set operations|

* Algebraic interior (core)
* Bounding points
* Convex hull
* Extreme point
* Interior
* Minkowski addition
* Polar

Template:Banach spaces Template:Lp spaces

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