In what follows, H will always denote a Hilbert space equipped with the scalar product( , )and the corresponding norm| |.
5.1 The parallelogram law.
5.4 Exercises for Chapter 5 147 SupposeEis a vector space equipped with a norm satisfying the parallelogram law, i.e.,
a+b2+ a−b2=2(a2+ b2) ∀a, b∈E.
Our purpose is to show that the quantity defined by (u, v)= 1
2(u+v2− u2− v2) u, v∈E, is a scalar product such that(u, u)= u2.
1. Check that
(u, v)=(v, u), (−u, v)= −(u, v)and(u,2v)=2(u, v) ∀u, v∈E.
2. Prove that
(u+v, w)=(u, w)+(v, w) ∀u, v, w∈E.
[Hint: Use the parallelogram law successively with (i)a =u, b =v; (ii)a = u+w, b=v+w, and (iii)a=u+v+w, b=w.]
3. Prove that(λu, v)=λ(u, v)∀λ∈R,∀u, v∈E.
[Hint: Consider first the caseλ∈N, thenλ∈Q, and finallyλ∈R.]
4. Conclude.
5.2 Lpis not a Hilbert space forp =2.
Letbe a measure space and assume that there exists a measurable setA⊂ such that 0<|A|<||.
Prove that the pnorm does not satisfy the parallelogram law for any 1≤p≤
∞,p =2.
[Hint: Use functions with disjoint supports.]
5.3 Let(un)be a sequence inHand let(tn)be a sequence in(0,∞)such that (tnun−tmum, un−um)≤0 ∀m, n.
1. Assume that the sequence(tn)isnondecreasing (possibly unbounded). Prove that the sequence(un)converges.
[Hint: Show that the sequence(|un|)is nonincreasing.]
2. Assume that the sequence(tn)isnonincreasing. Prove that the following alter- native holds:
(i) either|un| → ∞, (ii) or(un)converges.
Iftn →t >0, prove that(un)converges, and iftn →0, prove that both cases (i) and (ii) may occur.
5.4 LetK⊂H be a nonempty closed convex set. Letf ∈H and letu=PKf. Prove that
|v−u|2≤ |v−f|2− |u−f|2 ∀v∈K.
Deduce that
|v−u| ≤ |v−f| ∀v∈K.
Give a geometric interpretation.
5.5
1. Let(Kn)be a nonincreasingsequence of closed convex sets in H such that
∩nKn = ∅.
Prove that for everyf ∈Hthe sequenceun=PKnf converges (strongly) to a limit and identify the limit.
2. Let(Kn)be anondecreasingsequence of nonempty closed convex sets inH.
Prove that for everyf ∈ H the sequenceun =PKnf converges (strongly) to a limit and identify the limit.
Letϕ:H →Rbe a continuous function that is bounded from below. Prove that the sequenceαn=infKnϕconverges and identify the limit.
5.6 The radial projection onto the unit ball.
LetEbe a vector space equipped with the norm . Set
T u=
u ifu ≤1, u/u ifu>1.
1. Prove thatT u−T v ≤2u−v ∀u, v∈E.
2. Show that in general, the constant 2 cannot be improved.
[Hint: TakeE=R2with the normu = |u1| + |u2|.]
3. What happens if is a Hilbert norm?
5.7 Projection onto a convex cone.
LetK⊂H be a convex cone with vertex at 0, i.e.,
0∈K and λu+μv∈K ∀λ, μ >0, ∀u, v∈K; assume in addition thatKis closed.
Givenf ∈H, prove thatu=PKf ischaracterizedby the following properties:
u∈K, (f−u, v)≤0 ∀v∈K and (f−u, u)=0.
5.8 Letbe a measure space and leth:→ [0,+∞)be a measurable function.
Let
K= {u∈L2(); |u(x)| ≤h(x)a.e. on}.
Check thatKis a nonempty closed convex set inH=L2(). DeterminePK. 5.9 LetA⊂HandB ⊂Hbe two nonempty closed convex set such thatA∩B= ∅ andBis bounded.
5.4 Exercises for Chapter 5 149 Set
C=A−B.
1. Show thatCis closed and convex.
2. Setu=PC0 and writeu=a0−b0for somea0∈Aandb0∈B(this is possible sinceu∈C).
Prove that|a0−b0| =dist(A, B)=infa∈A,b∈B|a−b|.
DeterminePAb0andPBa0.
3. Supposea1 ∈Aandb1 ∈B is another pair such that|a1−b1| = dist(A, B).
Prove thatu=a1−b1.
Draw some pictures where the pair[a0, b0]is unique (resp. nonunique).
4. Find a simple proof of the Hahn–Banach theorem, second geometric form, in the case of a Hilbert space.
5.10 LetF :H →Rbe a convex function of classC1. LetK⊂Hbe convex and letu∈H. Show that the following properties are equivalent:
(i) F (u)≤F (v) ∀v∈K, (ii) (F(u), v−u)≥0 ∀v∈K.
Example:F (v)= |v−f|2withf ∈Hgiven.
5.11 Let M ⊂ H be a closed linear subspace that is not reduced to {0}. Let f ∈H, f /∈M⊥.
1. Prove that
m= inf
u∈M
|u|=1
(f, u) is uniquely achieved.
2. Let ϕ1, ϕ2, ϕ3 ∈ H be given and let E denote the linear space spanned by {ϕ1, ϕ2, ϕ3}. Determinemin the following cases:
(i) M=E, (ii) M=E⊥.
3. Examine the case in whichH=L2(0,1), ϕ1(t )=t, ϕ2(t )=t2, andϕ3(t )=t3. 5.12 Completion of a pre-Hilbert space.
LetE be a vector space equipped with the scalar product( , ). One doesnot assume thatEis complete for the norm|u| =(u, u)1/2(Eis said to be a pre-Hilbert space).
Recall that the dual spaceE, equipped with the dual normfE, is complete.
LetT :E→Ebe the map defined by
T u, vE,E =(u, v) ∀u, v∈E.
Check thatT is a linear isometry. IsT surjective?
Our purpose is to show thatR(T )is dense inEand that Eis a Hilbert norm.
1. Transfer toR(T )the scalar product ofEand extend it toR(T ). The resulting scalar product is denoted by((f, g))withf, g∈R(T ).
Check that the corresponding norm((f, f ))1/2coincides onR(T )withfE. Prove that
f, v =((f, T v)) ∀v∈E, ∀f ∈R(T ).
2. Prove thatR(T )=E.
[Hint: Givenf ∈ E, transferf to a linear functional on R(T ) and use the Riesz–Fréchet representation theorem inR(T ).]
Deduce thatEis a Hilbert space for the norm E.
3. Conclude that the completion ofEcan be identified withE. (For the definition of the completion see, e.g., A. Friedman [3].)
5.13 LetE be a vector space equipped with the norm E. The dual norm is denoted by E. Recall that the (multivalued) duality map is defined by
F (u)= {f ∈E; fE = uE andf, u = u2E}. 1. Assume thatF satisfies the following property:
F (u)+F (v)⊂F (u+v) ∀u, v∈E.
Prove that the norm Earises from a scalar product.
[Hint: Use Exercise 5.1.]
2. Conversely, if the norm E arises from a scalar product, what can one say aboutF?
[Hint: Use Exercise 5.12 and 1.1.]
5.14 Leta:H×H→Rbe a bilinear continuous form such that a(v, v)≥0 ∀v∈H.
Prove that the functionv →F (v)=a(v, v)is convex, of classC1, and determine its differential.
5.15 LetG⊂Hbe a linear subspace of a Hilbert spaceH;Gis equipped with the norm ofH. LetF be a Banach space. LetS:G→F be a bounded linear operator.
Prove that there exists a bounded linear operatorT :H →F that extendsSand such that
T
L(H,F )=S
L(G,F ). 5.16 The tripletV ⊂H ⊂V.
LetH be a Hilbert space equipped with the scalar product( , )and the corre- sponding norm| |. LetV ⊂H be a linear subspace that is dense inV. Assume that V has its own norm and thatV is a Banach space for . Assume also that the injectionV ⊂ H is continuous, i.e.,|v| ≤ Cv ∀v ∈ V. Consider the operator
5.4 Exercises for Chapter 5 151 T :H →Vdefined by
T u, vV,V =(u, v) ∀u∈H, ∀v ∈V . 1. Prove thatT uV ≤C|u| ∀u∈H.
2. Prove thatT is injective.
3. Prove thatR(T )is dense inVifV is reflexive.
4. Givenf ∈ V, prove thatf ∈ R(T ) iff there is a constanta ≥ 0 such that
|f, vV,V| ≤a|v| ∀v∈V.
5.17 LetM, N ⊂Hbe two closed linear subspaces.
Assume that(u, v)=0∀u∈M,∀v∈N. Prove thatM+Nis closed.
5.18 LetEbe a Banach space and letH be a Hilbert space. LetT ∈ L(E, H ).
Show that the following properties are equivalent:
(i) T admits a left inverse,
(ii) there exists a constantCsuch thatu ≤C|T u| ∀u∈E.
5.19 Let (un) be a sequence in H such that un u weakly. Assume that lim sup|un| ≤ |u|. Prove thatun→ustrongly without relying on Proposition 3.32.
5.20 Assume thatS∈L(H )satisfies(Su, u)≥0∀u∈H.
1. Prove thatN (S)=R(S)⊥.
2. Prove thatI+t Sis bijective for everyt >0.
3. Prove that
t→+∞lim (I+t S)−1f =PN (S)f ∀f ∈H.
[Hint: Two methods are possible:
(a) Consider the casesf ∈N (S)andf ∈R(S).
(b) Use weak convergence.]
5.21 Iterates of linear contractions. The ergodic theorem of Kakutani–Yosida.
LetT ∈ L(H )be such that T ≤ 1. Given f ∈ H and given an integer n≥1, set
σn(f )= 1
n(f +Tf +T2f + · · · +Tn−1f ) and
μn(f )= I+T
2 n
f.
Our purpose is to show that
nlim→∞σn(f )= lim
n→∞μn(f )=PN (I−T )f.
1. Check thatN (I−T )=R(I−T )⊥.
2. Assume thatf ∈R(I−T ). Prove that there exists a constantCsuch that|σn(f )| ≤ C/n∀n≥1.
3. Deduce that for everyf ∈H, one has
nlim→∞σn(f )=PN (I−T )f.
4. SetS =1
2(I+T ). Prove that
(1) |u−Su|2+ |Su|2≤ |u|2 ∀u∈H.
Deduce that
∞ i=0
|Siu−Si+1u|2≤ |u|2 ∀u∈H and that
|Sn(u−Su)| ≤ √|u|
n+1 ∀u∈H ∀n≥1.
5. Assume that f ∈ R(I −T ). Prove that there exists a constant C such that
|μn(f )| ≤C/√
n∀n≥1.
6. Deduce that for everyf ∈H, one has
nlim→∞μn(f )=PN (I−T )f.
5.22 LetC ⊂ H be a nonempty closed convex set and let T : C → C be a nonlinear contraction, i.e.,
|T u−T v| ≤ |u−v| ∀u, v∈C.
1. Let(un)be a sequence inCsuch that
un uweakly and(un−T un)→f strongly.
Prove thatu−T u=f.
[Hint: Start with the caseC=Hand use the inequality((u−T u)−(v−T v), u− v)≥0∀u, v.]
2. Deduce that ifCis bounded andT (C)⊂C, thenT has a fixed point.
[Hint: ConsiderTεu=(1−ε)T u+εawitha∈Cbeing fixed andε >0, ε→0.]
5.23 Zarantonello’s inequality.
LetT :H → H be a (nonlinear) contraction. Assume thatα1, α2, . . . , αn ∈ R are such thatαi ≥0∀iandn
i=1αi =1. Assume thatu1, u2, . . . , un∈Hand set σ =
n i=1
αiui.
5.4 Exercises for Chapter 5 153 Prove that
T σ − n
i=1
αiT ui 2≤ 1
2 n i,j=1
αiαj -
|ui−uj|2− |T ui−T uj|2 .
.
[Hint: Write T σ−
n i=1
αiT ui
2= n i,j=1
αiαj(T σ −T ui, T σ −T uj)
and use the identity(a, b)= 12(|a|2+ |b|2− |a−b|2).]
What can one deduce whenT is an isometry (i.e.,|T u−T v| = |u−v| ∀u, v∈H)?
5.24 The Banach–Saks property.
1. Assume that(un)is a sequence inH such thatun 0 weakly. Construct by induction a subsequence(unj)such thatun1 =u1and
|(unj, unk)| ≤ 1
k ∀k≥2 and∀j =1,2, . . . , k−1.
Deduce that the sequence(σp)defined byσp= 1pp
j=1unj converges strongly to 0 asp→ ∞.
[Hint: Estimate|σp|2.]
2. Assume that(un)is a bounded sequence inH. Prove that there exists a subse- quence(unj)such that the sequenceσp = p1p
j=1unj converges strongly to a limit asp→ ∞.
Compare with Corollary 3.8 and Exercise 3.4.
5.25 Variations on Opial’s lemma.
LetK⊂Hbe a nonempty closed convex set. Let(un)be a sequence inHsuch that foreachv∈Kthe sequence(|un−v|)is nonincreasing.
1. Check that the sequence(dist(un, K))is nonincreasing.
2. Prove that the sequence(PKun)converges strongly to a limit, denoted by. [Hint: Use Exercise 5.4.]
3. Assume here that the sequence(un)satisfies the property (P)
Whenever a subsequence(unk)converges weakly to some limitu∈H, thenu∈K.
Prove thatun weakly.
4. Assume here that
λ>0λ(K−K)=H. Prove that there exists someu∈Hsuch thatun uweakly andPKu=.
5. Assume here that IntK = ∅. Prove that there exists someu∈Hsuch thatun→u strongly.
[Hint: Consider first the case thatKis the unit ball and then the general case.]
6. Setσn = 1n(u1+u2+ · · · +un)and assume that the sequence(σn)satisfies property (P). Prove thatσn weakly.
5.26 Assume that(en)is an orthonormal basis ofH. 1. Check thaten0 weakly.
Let(an)be a bounded sequence inRand setun= 1nn
i=1aiei. 2. Prove that|un| →0.
3. Prove that√
n un0 weakly.
5.27 LetD⊂Hbe a subset such that the linear space spanned byDis dense inH. Let(En)n≥1be a sequence of closed subspaces inH that are mutually orthogonal.
Assume that
∞ n=1
|PEnu|2= |u|2 ∀u∈D.
Prove thatH is the Hilbert sum of theEn’s.
5.28 Assume thatHis separable.
1. LetV ⊂ H be a linear subspace that is dense inH. Prove thatV contains an orthonormal basis ofH.
2. Let(en)n≥1be an orthonormal sequence inH, i.e.,(ei, ej)=δij. Prove that there exists an orthonormal basis ofH that contains∞
n=1{en}. 5.29 A lemma of Grothendieck.
Letbe a measure space with||<∞. LetEbe a closed subspace ofLp()with 1≤p <∞. Assume thatE⊂L∞(). Our purpose is to prove that dimE <∞.
1. Prove that there exists a constantCsuch that
u∞≤Cup ∀u∈E.
[Hint: Use Corollary 2.8.]
2. Prove that there exists a constantMsuch that
u∞≤Mu2 ∀u∈E.
[Hint: Distinguish the cases 1≤p≤2 and 2< p <∞.]
3. Deduce thatEis a closed subspace ofL2().
In what follows we assume that dimE = ∞. Let(en)n≥1 be an orthonormal sequence ofE(equipped with theL2scalar product).
4. Fix any integerk≥1. Prove that there exists a null setω⊂such that k
i=1
αiei(x)≤M k
i=1
αi2 1/2
∀x∈\ω, ∀α=(α1, α2, . . . , αk)∈Rk.
5.4 Exercises for Chapter 5 155 [Hint: Start with the caseα∈Qk.]
5. Deduce thatk
i=1|ei(x)|2≤M2∀x∈\ω.
6. Conclude.
5.30 Let(en)n≥1be an orthonormalsequenceinH =L2(0,1). Letp(t )be a given function inH.
1. Prove that for everyt ∈ [0,1], one has (1)
∞ n=1
t 0
p(s)en(s)ds 2≤
t
0
|p(s)|2ds.
2. Deduce that (2)
∞ n=1
1 0
t 0
p(s)en(s)ds 2dt≤
1 0
|p(t )|2(1−t )dt.
3. Assume now that(en)n≥1is an orthonormalbasisofH. Prove that (1) and (2) become equalities.
4. Conversely, assume that equality holds in (2) and thatp(t ) = 0 a.e. Prove that (en)n≥1is an orthonormal basis.
Example:p≡1.
5.31 The Haar basis.
Given an integern≥1, writen=k+2p, wherep ≥0 andk≥0 are integers uniquely determined by the conditionk≤2p−1. Consider the function defined on (0,1)by
ϕn(t )=
⎧⎪
⎪⎪
⎨
⎪⎪
⎪⎩
2p/2 if k2−p< t < (k+1 2)2−p,
−2p/2 if (k+1
2)2−p< t < (k+1)2−p,
0 elsewhere.
Setϕ0≡1 and prove that(ϕn)n≥0is an orthonormal basis ofL2(0,1).
5.32 The Rademacher system and the Walsh basis.
For every integeri ≥ 0 consider the functionri(t )defined on(0,1)byri(t )= (−1)[2it](as usual[x]denotes the largest integer≤x).
1. Check that(ri)i≥0is an orthonormal sequence inL2(0,1)(called the Rademacher system).
2. Is(ri)i≥0an orthonormal basis?
[Hint: Consider the functionu=r1r2.]
3. Given an integern≥ 0, consider its binary representationn=
i=0αi2i with αi ∈ {0,1}.
Set
wn(t )=
&
i=0
ri+1(t )αi.
Prove that(wn)n≥0is an orthonormal basis ofL2(0,1)(called the Walsh basis).
Note that(ri)i≥0is a subset of(wn)n≥0.