1.1 Properties of the duality map.
LetEbe an n.v.s. The duality mapF is defined for everyx∈Eby F (x)= {f ∈E; f = xandf, x = x2}. 1. Prove that
F (x)= {f ∈E; f ≤ xandf, x = x2} and deduce thatF (x)is nonempty, closed, and convex.
2. Prove that ifEis strictly convex, thenF (x)contains a single point.
3. Prove that F (x)=
f ∈E; 1
2y2−1
2x2≥ f, y−x ∀y∈E
. 4. Deduce that
F (x)−F (y), x−y ≥0 ∀x, y ∈E,
and more precisely that
f −g, x−y ≥0 ∀x, y∈E, ∀f ∈F (x), ∀g∈F (y).
Show that, in fact,
f −g, x−y ≥(x − y)2 ∀x, y ∈E, ∀f ∈F (x), ∀g∈F (y).
5. Assume again thatEis strictly convex and letx, y∈Ebe such that F (x)−F (y), x−y =0.
Show thatF x=F y.
1.2 LetEbe a vector space of dimensionnand let(ei)1≤i≤nbe a basis ofE. Given x∈E, writex =n
i=1xiei withxi ∈R; givenf ∈E, setfi = f, ei. 1. Consider onEthe norm
x1= n i=1
|xi|.
(a) Compute explicitly, in terms of thefi’s, the dual normfE off ∈E. (b) Determine explicitly the setF (x)(duality map) for everyx ∈E.
2. Same questions but whereEis provided with the norm x∞= max
1≤i≤n|xi|. 3. Same questions but whereEis provided with the norm
x2= n
i=1
|xi|2 1/2
,
and more generally with the norm xp =
n
i=1
|xi|p 1/p
, wherep∈(1,∞).
1.3 LetE= {u∈C([0,1];R);u(0)=0}with its usual norm u = max
t∈[0,1]|u(t )|. Consider the linear functional
1.4 Exercises for Chapter 1 21 f :u∈E→f (u)=
1 0
u(t )dt.
1. Show thatf ∈Eand computefE.
2. Can one find someu∈Esuch thatu =1 andf (u)= fE?
1.4 Consider the spaceE = c0(sequences tending to zero) with its usual norm (see Section 11.3). For every elementu=(u1,u2,u3, . . . )inEdefine
f (u)= ∞ n=1
1 2nun.
1. Check thatf is a continuous linear functional onEand computefE. 2. Can one find someu∈Esuch thatu =1 andf (u)= fE?
1.5 LetEbe an infinite-dimensional n.v.s.
1. Prove (using Zorn’s lemma) that there exists an algebraic basis(ei)iεI inEsuch thatei =1∀i∈I.
Recall that an algebraic basis (or Hamel basis) is a subset(ei)iεI inEsuch that everyx∈Emay be written uniquely as
x =
iεJ
xiei withJ ⊂I, J finite.
2. Construct a linear functionalf :E→Rthat is not continuous.
3. Assuming in addition thatEis a Banach space, prove thatI is not countable.
[Hint:Use Baire category theorem (Theorem 2.1).]
1.6 LetEbe an n.v.s. and letH ⊂Ebe a hyperplane. LetV ⊂Ebe an affine subspace containingH.
1. Prove that eitherV =H orV =E.
2. Deduce thatHis either closed or dense inE.
1.7 LetEbe an n.v.s. and letC ⊂Ebe convex.
1. Prove thatCand IntCare convex.
2. Givenx ∈Candy∈IntC, show thatt x+(1−t )y∈IntC ∀t ∈(0,1).
3. Deduce thatC =IntCwhenever IntC = ∅.
1.8 LetEbe an n.v.s. with norm . LetC⊂Ebe an open convex set such that 0∈C. Letpdenote the gauge ofC(see Lemma 1.2).
1. AssumingC is symmetric (i.e.,−C = C)andC is bounded, prove thatpis a norm which is equivalent to .
2. LetE=C([0,1]; R)with its usual norm u = max
t∈[0,1]|u(t )|. Let
C=
u∈E; 1
0
|u(t )|2dt <1
.
Check thatC is convex and symmetric and that 0 ∈ C. Is C bounded inE?
Compute the gaugepofCand show thatpis a norm onE. Ispequivalent to ?
1.9 Hahn–Banach in finite-dimensional spaces.
LetEbe a finite-dimensional normed space. LetC ⊂Ebe a nonempty convex set such that 0∈/C. We claim that there always exists some hyperplane that separates Cand{0}.
[Note that every hyperplane is closed (why?). The main point in this exercise is that no additional assumption onCis required.]
1. Let(xn)n≥1be a countable subset ofC that is dense inC (why does it exist?).
For everynlet
Cn=conv{x1, x2, . . . , xn} =
x= n i=1
tixi; ti ≥0∀iand n i=1
ti =1
.
Check thatCnis compact and that∞
n=1Cnis dense inC.
2. Prove that there is somefn∈Esuch that
fn =1 andfn, x ≥0 ∀x∈Cn. 3. Deduce that there is somef ∈Esuch that
f =1 andf, x ≥0 ∀x ∈C.
Conclude.
4. LetA, B ⊂ Ebe nonempty disjoint convex sets. Prove that there exists some hyperplaneHthat separatesAandB.
1.10 LetEbe an n.v.s. and letI be any set of indices. Fix a subset(xi)iεI inEand a subset(αi)iεI inR. Show that the following properties are equivalent:
There exists somef ∈Esuch thatf, xi =αi ∀i∈I .
(A) ⎧
⎪⎪
⎨
⎪⎪
⎩
There exists a constantM≥0 such that for each finite subset J ⊂I and for every choice of real numbers(βi)i∈J, we have
i∈J
βiαi≤M
i∈J
βixi. (B)
1.4 Exercises for Chapter 1 23 Note that in the proof of (B)⇒(A) one may find somef ∈EwithfE≤M.
[Hint:Try first to definef on the linear space spanned by the(xi)iεI.]
1.11 LetEbe an n.v.s. and letM >0. Fixnelements(f1)1≤i≤ninEandnreal numbers(αi)1≤i≤n. Prove that the following properties are equivalent:
∀ε >0 ∃xε ∈Esuch that
xε ≤M+εandfi, xε =αi ∀i=1,2, . . . , n.
(A)
n
i=1
βiαi≤M n
i=1
βifi ∀β1, β2, . . . , βn∈R. (B)
[Hint:For the proof of (B)⇒(A) consider first the case in which thefi’s are linearly independent and imitate the proof of Lemma 3.3.]
Compare Exercises 1.10, 1.11 and Lemma 3.3.
1.12 LetEbe a vector space. Fixnlinear functionals(fi)1≤i≤n onEandnreal numbers(αi)1≤i≤n. Prove that the following properties are equivalent:
There exists somex ∈Esuch thatfi(x)=αi ∀i=1,2, . . . , n.
(A)
For any choice of real numbersβ1, β2, . . . , βnsuch that n
i=1βifi =0, one also hasn
i=1βiαi =0.
(B)
1.13 LetE=Rnand let
P = {x∈Rn; xi ≥0 ∀i=1,2, . . . , n}.
LetMbe a linear subspace ofEsuch thatM∩P = {0}. Prove that there is some hyperplaneHinEsuch that
M⊂HandH∩P = {0}. [Hint:Show first thatM⊥∩IntP = ∅.]
1.14 LetE=1(see Section 11.3) and consider the two sets X= {x =(xn)n≥1∈E; x2n=0∀n≥1}
and
Y =
y=(yn)n≥1∈E; y2n= 1
2ny2n−1∀n≥1
.
1. Check thatXandY are closed linear spaces and thatX+Y =E.
2. Letc∈Ebe defined by
c2n−1=0 ∀n≥1, c2n= 21n ∀n≥1.
Check thatc /∈X+Y.
3. SetZ=X−cand check thatY ∩Z = ∅. Does there exist a closed hyperplane inEthat separatesY andZ?
Compare with Theorem 1.7 and Exercise 1.9.
4. Same questions inE=p, 1< p <∞, and inE=c0.
1.15 LetEbe an n.v.s. and letC⊂Ebe a convex set such that 0∈C. Set C= {f ∈E; f, x ≤1 ∀x ∈C},
(A)
C= {x ∈E; f, x ≤1 ∀f ∈C}. (B)
1. Prove thatC=C.
2. What isCifCis a linear space?
1.16 LetE=1, so thatE =∞(see Section 11.3). ConsiderN =c0as a closed subspace ofE.
Determine
N⊥= {x ∈E; f, x =0 ∀f ∈N} and
N⊥⊥= {f ∈E; f, x =0 ∀x ∈N⊥}. Check thatN⊥⊥ =N.
1.17 LetEbe an n.v.s. and letf ∈ E withf = 0. Let M be the hyperplane [f =0].
1. DetermineM⊥.
2. Prove that for everyx ∈E, dist(x, M)=infy∈Mx−y = |f,xf|. [Find a direct method or use Example 3 in Section 1.4.]
3. Assume now thatE= {u∈C([0,1];R);u(0)=0}and that f, u =
1 0
u(t )dt, u∈E.
Prove that dist(u, M)= |1
0 u(t )dt| ∀u∈E.
Show that infv∈Mu−vis never achieved for anyu∈E\M.
1.18 Check that the functionsϕ : R → (−∞,+∞]defined below are convex l.s.c. and determine the conjugate functionsϕ. Draw their graphs and mark their epigraphs.
1.4 Exercises for Chapter 1 25 ϕ(x)=ax+b, wherea, b∈R.
(a)
ϕ(x)=ex. (b)
ϕ(x)=
0 if|x| ≤1, +∞ if|x|>1.
(c)
ϕ(x)=
0 ifx =0,
+∞ ifx =0.
(d)
ϕ(x)=
−logx ifx >0, +∞ ifx≤0.
(e)
ϕ(x)=
−(1−x2)1/2 if|x| ≤1, +∞ if|x|>1.
(f)
ϕ(x)= 1
2|x|2 if|x| ≤1,
|x| − 12 if|x|>1.
(g)
ϕ(x)= 1
p|x|p, where 1< p <∞. (h)
ϕ(x)=x+=max{x,0}. (i)
ϕ(x)= 1
pxp ifx≥0, where 1< p <+∞, +∞ ifx <0.
(j)
ϕ(x)=
−p1xp ifx ≥0, where 0< p <1, +∞ ifx <0.
(k)
ϕ(x)= 1
p[(|x| −1)+]p, where 1< p <∞. (l)
1.19 LetEbe an n.v.s.
1. Letϕ, ψ : E → (−∞,+∞]be two functions such that ϕ ≤ ψ. Prove that ψ≤ϕ.
2. LetF : R→ (−∞,+∞]be a convex l.s.c. function such thatF (0)= 0 and F (t )≥0∀t ∈R. Setϕ(x)=F (x).
Prove thatϕis convex l.s.c. and thatϕ(f )=F(f)∀f ∈E.
1.20 LetE =pwith 1≤ p <∞(see Section 11.3). Check that the functions ϕ : E → (−∞,+∞]defined below are convex l.s.c. and determineϕ. Forx = (x1, x2, . . . , xn, . . . )set
ϕ(x)= +∞
k=1 k|xk|2 if∞
k=1 k|xk|2<+∞,
+∞ otherwise.
(a)
ϕ(x)=
+∞
k=2
|xk|k. (Check thatϕ(x) <∞for everyx ∈E.) (b)
ϕ(x)=
⎧⎪
⎨
⎪⎩
+∞
k=1
|xk| if ∞ k=1
|xk|<+∞,
+∞ otherwise.
(c)
1.21 LetE=E=R2and let
C= {[x1, x2]; x1≥0, x2≥0}. OnEdefine the function
ϕ(x)= −√
x1x2 ifx ∈C, +∞ ifx /∈C.
1. Prove thatϕis convex l.s.c. onE.
2. Determineϕ.
3. Consider the setD= {[x1, x2];x1=0}and the functionψ=ID. Compute the value of the expressions
xinf∈E{ϕ(x)+ψ (x)} and sup
f∈E
{−ϕ(−f )−ψ(f )}.
4. Compare with the conclusion of Theorem 1.12 and explain the difference.
1.22 LetEbe an n.v.s. and letA⊂Ebe a closed nonempty set. Let ϕ(x)=dist(x, A)= inf
a∈Ax−a. 1. Check that|ϕ(x)−ϕ(y)| ≤ x−y ∀x, y ∈E.
2. Assuming thatAis convex, prove thatϕis convex.
3. Conversely, assuming thatϕis convex, prove thatAis convex.
4. Prove thatϕ=(IA)+IBE for everyAnot necessarily convex.
1.23 Inf-convolution.
LetEbe an n.v.s. Given two functionsϕ, ψ:E→(−∞,+∞], one defines the inf-convolutionofϕandψas follows: for everyx ∈E, let
(ϕ∇ψ )(x)= inf
y∈E{ϕ(x−y)+ψ (y)}. Note the following:
(i) (ϕ∇ψ )(x)may take the values±∞, (ii) (ϕ∇ψ )(x) <+∞iffx ∈D(ϕ)+D(ψ ).
1. Assuming thatD(ϕ)∩D(ψ) = ∅, prove that(ϕ∇ψ )does not take the value
−∞and that
1.4 Exercises for Chapter 1 27 (ϕ∇ψ )=ϕ+ψ.
2. Assuming thatD(ϕ)∩D(ψ ) = ∅, prove that
(ϕ+ψ ) ≤(ϕ∇ψ)onE.
3. Assume thatϕandψare convex and there existsx0∈D(ϕ)∩D(ψ )such thatϕ is continuous atx0. Prove that
(ϕ+ψ )=(ϕ∇ψ)onE.
4. Assume thatϕ andψare convex and l.s.c., and thatD(ϕ)∩D(ψ ) = ∅. Prove that
(ϕ∇ψ)=(ϕ+ψ )onE.
Given a functionϕ :E→(−∞,+∞], set
epistϕ= {[x, λ] ∈E×R; ϕ(x) < λ}. 5. Check thatϕis convex iff epistϕis a convex subset ofE×R.
6. Letϕ, ψ :E→(−∞,+∞]be functions such thatD(ϕ)∩D(ψ) = ∅. Prove that
epist(ϕ∇ψ )=(epistϕ)+(epistψ ).
7. Deduce that ifϕ, ψ :E→(−∞,+∞]are convex functions such thatD(ϕ)∩ D(ψ) = ∅, then(ϕ∇ψ )is a convex function.
1.24 Regularization by inf-convolution.
LetEbe an n.v.s. and letϕ :E→(−∞,+∞]be a convex l.s.c. function such thatϕ ≡ +∞. Our aim is to construct a sequence of functions(ϕn)such that we have the following:
(i) For everyn,ϕn:E→(−∞,+∞)is convex and continuous.
(ii) For everyx, the sequence(ϕn(x))nis nondecreasing and converges toϕ(x).
For this purpose, let
ϕn(x)= inf
y∈E{nx−y +ϕ(y)}.
1. Prove that there is someN, large enough, such that forn≥N,ϕn(x)is finite for allx ∈E. From now on, one choosesn≥N.
2. Prove thatϕnis convex (see Exercise 1.23) and that
|ϕn(x1)−ϕn(x2)| ≤nx1−x2 ∀x1, x2∈E.
3. Determine(ϕn).
4. Check thatϕn(x)≤ϕ(x) ∀x ∈E,∀n. Prove that for everyx ∈E, the sequence (ϕn(x))nis nondecreasing.
5. Givenx ∈D(ϕ), chooseyn∈Esuch that
ϕn(x)≤nx−yn +ϕ(yn)≤ϕn(x)+1 n. Prove that limn→∞yn=xand deduce that limn→∞ϕn(x)=ϕ(x).
6. Forx /∈D(ϕ), prove that limn→∞ϕn(x)= +∞. [Hint:Argue by contradiction.]
1.25 A semiscalar product.
LetEbe an n.v.s.
1. Letϕ :E→(−∞,+∞)be convex. Givenx, y ∈E, consider the function h(t )=ϕ(x+ty)−ϕ(x)
t , t >0.
Check thathis nondecreasing on(0,+∞)and deduce that limt↓0h(t )=inf
t >0h(t )exists in[−∞,+∞).
Define the semiscalar product[x, y]by [x, y] =inf
t >0
1
2t[x+ty2− x2]. 2. Prove that|[x, y]| ≤ xy ∀x, y∈E.
3. Prove that
[x, λx+μy] =λx2+μ[x, y] ∀x, y∈E, ∀λ∈R, ∀μ≥0 and
[λx, μy] =λμ[x, y] ∀x, y ∈E, ∀λ≥0, ∀μ≥0.
4. Prove that for everyx ∈ E, the functiony → [x, y]is convex. Prove that the functionG(x, y)= −[x, y]is l.s.c. onE×E.
5. Prove that
[x, y] = max
f∈F (x)f, y ∀x, y ∈E,
whereF denotes the duality map (see Remark 2 following Corollary 1.3 and Exercise 1.1).
[Hint:Setα= [x, y]and apply Theorem 1.12 to the functionsϕandψdefined as follows:
ϕ(z)=1
2x+z2−1
2x2, z∈E, and
ψ (z)=
−t α whenz=tyandt ≥0, +∞ otherwise.]
1.4 Exercises for Chapter 1 29 6. Determine explicitly[x, y], whereE =Rn with the normxp, 1≤ p ≤ ∞
(see Section 11.3).
[Hint:Use the results of Exercise 1.2.]
1.26 Strictly convex norms and functions.
LetEbe an n.v.s. One says that thenorm isstrictly convex(or that thespace Eisstrictly convex) if
t x+(1−t )y<1, ∀x, y∈Ewithx =y, x = y =1, ∀t∈(0,1).
One says that afunctionϕ :E→(−∞,+∞]isstrictly convexif
ϕ(t x+(1−t )y) < t ϕ(x)+(1−t )ϕ(y) ∀x, y∈Ewithx =y, ∀t ∈(0,1).
1. Prove that thenorm is strictly convex iff thefunctionϕ(x)= x2is strictly convex.
2. Same question withϕ(x)= xpand 1< p <∞.
1.27 LetEandF be two Banach spaces and letG ⊂ E be a closed subspace.
LetT :G→F be a continuous linear map. The aim is to show that sometimes,T cannot be extended by a continuous linear mapT:E→F. For this purpose, letE be a Banach space and letG⊂Ebe a closed subspace that admits no complement (see Remark 8 in Chapter 2). LetF =GandT =I (the identity map). Prove that T cannot be extended.
[Hint:Argue by contradiction.]
Compare with the conclusion of Corollary 1.2.