Functional Analysis

Functional Analysis

(collection of some definitions and results)

0. Prerequsites

0.1. Differential and integral calculus of one and several variables 0.2. Linear algebra: vector spaces, linear maps, inner products, dual space 0.3. Measure theory: null sets, equality “almost everywhere”, integrable functions 0.4. General topology: compactness, continuity, density

0.5.All norms in finite dimension are equivalent

1. Basic integration results

1.1.Definition.

A function f :Ω→Risintegrableif^R_Ω|f|<∞. The set of all integrable functions onΩ is denotedL¹(Ω). Two functions ofL¹(Ω)that coincide almost everywhere are considered as the same function. Using the normkfk_L1(Ω):=^R_Ω|f|, this set is a normed vector space (and it turns out to be complete).

1.2.Theorem (monotone convergence, Beppo Levi)

Let f_nan increasing sequence of positiveL¹functions such that sup_n^R f_n<∞. Then f_n(x)con- verges almost everywhere to a finite limit denoted f(x). Moreover, f ∈L¹(Ω)andkf_n−fk_L1→ 0.

1.3.Theorem (dominated convergence, Lebesgue)

Let f_na sequence ofL¹functions converging pointwise a.e. to a function f(x). Assume there exist a functionh∈L¹(Ω)such that|f_n(x)| ≤h(x)almost everywhere. Thenf∈L¹(Ω)andkf_n−

fk_L1→0.

1.4.Theorem (density)

The setC_c(Ω)of compactly supported continuous functions is dense inL¹(Ω).

1.5.Theorem (Tonelli)

If f :Ω1×Ω2→R⁺∪ {+∞}then ZZ

Ω1×Ω₂

f(x,y)dxdy = Z

Ω1

Z Ω2

f(x,y)dy

dx = Z

Ω2

Z Ω1

f(x,y)dx

dy.

in particular, if one of these three integrals is+∞, then the other two also are also. (In words: the sum of positive numbers, be it finite or infinite, does not depend on the order in which they are summed.)

1.6.Theorem (Fubini)

If f ∈L¹(Ω₁×Ω₂)then the three members in the formula above are well-defined and they are equal. In particular, the fact that the second member is well defined implies that

- The functiony7→ f(x,y)belongs toL¹(Ω₂)for a.e.x - The functionx7→^R_Ω

2 f(x,y)dythus defined belongs toL¹(Ω₁).

2. L

spaces

2.1.Definition.Let us define the following numbers (which may be infinite) kfk_Lp(Ω):=

_Z

Ω

|f|^p 1/p

for 1≤p<∞

kfk_L∞(Ω):=inf

M ; |f(x)| ≤Ma.e.onΩ

The spaceL^p(Ω)for 1≤p≤∞is the set of functions such thatkfk_Lp(Ω)<∞.

To eachp∈[1,∞]we associate itsconjugate exponent p⁰such that¹_p+_p¹0 =1.

2.2.H¨older inequality.Letp∈[1,∞], f ∈L^pandg∈L^p0. Then f g∈L¹and Z

|f g| ≤ kfk

L^pkgk_Lp⁰

2.3.Minkowski’s inequality.Letp∈[1,∞]and f,g∈L^p. Then kf+gk_L^p ≤ kfk_L^p + kgk_L^p

2.4.Interpolation inequality.Let 1≤s≤r≤t≤∞and f∈L^s∩L^t. Then f ∈L^rand kfk_Lr ≤ kfk^θ_Lskgk^1−θ_Lt

whereθ=^s(t−r)_r(t−s).

2.6.Theorem (Fischer-Riesz).L^pis a complete normed vector space forp∈[1,∞].

2.7. Riesz representation theorem. Let p∈[1,∞)and letϕ:L^p→Rbe a linear continuous function. Then there exist a uniqueu∈L^p0 such that

ϕ(f) = Z

u f ∀f ∈L^p

Thus, the topological dual ofL^pisL^p0.

2.8.Definition.L_loc^p (Ω):= functions such that fχK∈L^p(Ω)for all compactsK⊆Ω 2.9.Fundamental lemma.Let f ∈L¹_loc(Ω)such that

Z

f u=0 ∀u∈C_c(Ω) then f =0 almost everywhere onΩ.

2.9.Theorem (density).The spaceC_c(Ω)is dense inL^p(Ω)for 1≤p<∞.

2.10.Theorem (convolution inL^p).Theconvolutionof f,g:R^N→Ris defined by (f∗g)(x):=

Z

R

f(x−y)g(y)dy

It is well-defined in the following cases:

f g f∗g

L¹ L¹ L¹ L¹ L² L² L^p L^p0 L^∞∩C

3. Hilbert spaces

3.1. Definition. Areal Hilbert spaceis a real vector spaceHwith an inner product(f,g), such thatHis complete under the normkfk:=p

(f,f). Examples:R^NandL²(Ω).

3.2.Cauchy-Schwarz inequality.|(f,g)| ≤ kfk kgk

3.3.Pythagoras theorem.If(f,g) =0 thenkf+gk²=kfk²+kgk². If the sequence f_kis pairwise orthogonal thenk∑kf_kk²=∑kkf_kk². 3.4.Parallelogram identity.kfk² + kgk²=1

2 kf+gk² +kf−gk² 3.5.Polarization identity.(f,g) =1

4 kf+gk²− kf−gk²

3.6.Theorem (projection on a closed convex set).LetC⊆Hbe a nonempty closed convex set.

Then, for any f∈Hthere exist a uniqueg∈Cthat minimizes the distance to f. The pointgis characterized by the relation∀h∈C,(f−g,h−g)≤0.

3.7.Definition.LetA⊆H. TheorthogonalofAis the set A^⊥:={f,∀a∈A,(f,a) =0}

It is a closed subspace ofH.

3.8.Theorem (orthogonal decomposition).LetF⊆Hbe a closed subspace. Then any element ofHdecomposes in a unique way in the form f=g+hwhereg∈Fandh∈F^⊥. Moreover,g is the projection of f onFandhis the projection of f onF^⊥.

3.9. Theorem (Riesz). For any f ∈H, the mapu7→(u,f)is a continuous linear map fromH toR. Conversely, ifϕ :H→Ris a continuous linear map, there exists a unique f ∈H such thatϕ(u) = (u,f) ∀u∈H.

Thus, the dual ofHis isomorphic canonically toH.

3.10.Definition.AHilbert basisofHis an orthonormal sequence of vectorse_n,n∈Nwhich is total (it spans the whole spaceH).

3.11.Theorem.Any (separable) Hilbert space admits a Hilbert basis.

3.12. Theorem (“Fourier” series). Any element of f can be written in a unique way using a Hilbert basis

f =

∑

n

c_ne_n.

The coefficients arec_n= (f,e_n)and they satisfyParseval’s identitykfk²=∑nkc_nk². 3.13.Corollary.All (separable) Hilbert spaces are isomorphic (to`²(N)).

3.14.Continuity of linear forms. Letλ:H→Rbe a linear form. We say that is continuous if and only if there exists a constantCsuch that|λ(f)| ≤Ckfk.

3.15. Continuity of bilinear forms. Let a:H×H →Rbe a bilinear form. We say that is continuous if and only if there exists a constantCsuch that|a(f,g)| ≤Ckfkkgk. Moreover, if there exists a constantc>0 such thata(f,f)≥ckfk²then the form is said to becoercive.

3.16.Theorem (Lax-Milgram, symmetric case).LetEbe an energy of the form E(f) =1

2a(f,f)−b(f)

whereais a continuous bilinear, symmetric and coercive form, andbis a continuous linear form.

Then there is a unique f that minimizesE(f). Moreover, it is characterized by the relation a(f,u) =b(u) ∀u∈H.

4. Distributions

4.1. Definitions.LetΩ⊆Rbe a connected subset. We introduce the following spaces D(Ω):=C_c^∞(Ω)

D⁰(Ω):=dual ofD(Ω)

The dual is taken with respect to a topology of D defined elsewhere. The elements ofD⁰are calleddistributions.

4.2. Notation.IfT :D→Ris a distribution, we use the following notations forT(ϕ):

T(ϕ) = (T,ϕ) =

Z

Ω

Tϕ =

Z

Ω

T(x)ϕ(x)dx

4.3. Definition.We say that a sequence of distributionsT_nconverges toT if for any functionϕ∈ Dwe have(T_n,ϕ)→(T,ϕ).

4.4. Examples.

1) To any function f∈L¹_loc(Ω)we associate a distributionT_f defined by (T_f,ϕ):=

Z

Ω

f(x)ϕ(x)dx.

This association is injective, and denoted simply byT_f =f. 2) Dirac delta:(δ,ϕ):=ϕ(0)

3) Derivative of Dirac delta:(δ⁰,ϕ):=−ϕ⁰(0) 4) Dirac comb:(X,ϕ):=∑k∈Zϕ(2πk).

5) Principal value of 1/x:

(vp(1/x),ϕ):=lim

ε→0 Z _∞

ε

ϕ(x)−ϕ(−x)

x dx

or equivalently, vp(1/x):=ln(|x|)⁰(defined below) 4.5. Derivative of a distribution.(T⁰,ϕ):= (T,−ϕ⁰)

4.6. Proposition.This definition of the derivative coincides with the classical one: whenT =T_f for a differentiable f, thenT_f⁰=T_f⁰.

4.7. Product of a function and a distribution.(f T,ϕ):= (T,fϕ) Observation: the product of two distributions cannot be defined in general.

5. Sobolev spaces

5.1. Definition.We define the following spaces, for 1≤p≤∞andk=1,2,3, . . .

W^k,p(Ω) :=







functionsf :Ω→Rsuch that all their distributional derivatives of order 0, . . . ,kbelong toL^p(Ω)

)

H^k(Ω) := W^k,2(Ω).

The spaceW^k,p(Ω)is endowed with the following norm (written here for the 1D caseΩ⊆R):

kfk_k,p:=

kfk^p_p+kf⁰k^p_p+· · · kf^(k)k^p_p1/p

5.2. Properties.The spacesW^k,pare complete normed vector spaces. The spacesH^kare Hilbert spaces (thus, their norms can be computed using a Hilbert basis and Parseval’s identity).

6. Bibliography

6.1.H. Br´ezis :Analyse Fonctionnelle (canonical reference for functional analysis) 6.2.L. C. Evans :Partial Differential Equations (canonical reference for PDE)

6.3.C. Gasquet, P. Witomski :Analyse de Fourier et applications (nice exposition of the theory of distributions)

7. Exercices

E1. Find two counterexamples to Beppo Levi (missing the condition of monotonicity). One example using compactly-supported functions and another one using bounded functions.

E2. Find two counterexamples to the dominated convergence theorem (missing the dominat- ing function). One example using compactly-supported functions and another one using bounded functions.

*E3. Can you find a function f(x,y) that is integrable along each variable and^R(^R f(x,y)dx)dy=2 but^R(^R f(x,y)dy)dx=3.

E4. Find functions satisfying each of the following conditions (or prove that they do not exist) (a) f∈L¹⁰(R), f 6∈L¹²(R)

(b) f6∈L¹⁰(R), f ∈L¹²(R)

(c) f∈L¹⁰(R), f 6∈L¹²(R), f ∈L¹⁴(R) (d) f6∈L¹⁰(R), f ∈L¹²(R), f 6∈L¹⁴(R) E5. Prove that the dual ofL¹isL^∞.

**E6. Prove that the dual ofL^∞is notL¹.

E7. Find a Hilbert basis ofL²([0,T])consisting of sine and cosine functions.

E8. Find a Hilbert basis ofL²([0,T])consisting of only sine functions.

E9. Find a Hilbert basis ofL²([0,T])consisting of only cosine functions.

*E10. Find a Hilbert basis ofL²(R).

E11. The polarization identity allows to compute the scalar product from the associated norm.

Does it imply that any normed space has an associated scalar product?

E12. Prove proposition 4.6 above.

E13. Define the shift of a distribution onRby a lengtha. (Analogous to the shiftf(x−a)of a functionf).

E14. Ifτ_aT denotes the shift ofT, prove thatT⁰=limh→0T−τ_hT h

E15. Write down the scalar product that leads to the Sobolev norm on section 6.

E16. LetΩ⊆R²andf ∈L²(Ω). Prove that there is a unique functionuminimizing the energy E(u) =1

2 Z

k∇u(x,y)k²dxdy−

Z

f(x,y)u(x,y)dxdy

and satisfying the conditionu=0 on∂Ω. Moreover, the optimal function satisfies Poisson equation−∆u= f. (Hint: define an appropriate Hilbert space, and apply Lax-Milgram theorem.)