Part XII, Chapter A
Banach and Hilbert Spaces
The goal of this appendix is to recall fundamental results on Banach and
Hilbert spaces. The results collected herein provide a theoretical framework
for the mathematical analysis of the finite element method. Some classical
results are stated without proof; see Aubin [24], Brezis [97], Lax [321], Rudin
[399], Yosida [483], Zeidler [486] for further insight.
One important outcome of this appendix is the characterization of bijective
operators in Banach spaces. To get started, let us recall the following definition
of injective, surjective, and bijective maps.
Definition A.1 (Injection, surjection, bijection). Let E and F be two
sets. A function (or map) f : E −→ F is said to be injective if every element
of the codomain (i.e., F ) is mapped to by at most one element of the domain
(i.e., E). The function is said to be surjective if every element of the codomain
is mapped to by at least one element of the domain. Finally, f is bijective if
every element of the codomain is mapped to by exactly one element of the
domain (i.e., f is both injective and surjective).
A.1 Normed vector spaces
Definition A.2 (Norm). Let V be a vector space over the field K = R or C.
A norm on V is a map
k·kV : V ∋ v 7−→ kvkV ∈ [0, ∞),
satisfying the following three properties:
(i) Definiteness: kvkV = 0 ⇐⇒ v = 0.
(ii) 1-homogeneity: kλvkV = |λ| kvkV , for all λ ∈ K and all v ∈ V .
(iii) Triangle inequality: kv + wkV ≤ kvkV + kwkV , for all v, w ∈ V .
(A.1)
774
Appendix A. Banach and Hilbert Spaces
A seminorm on V is a map from V to [0, ∞) which satisfies only properties
(ii) and (iii).
Definition A.3 (Equivalent norms). Two norms k·kV,1 and k·kV,2 are said
to be equivalent on V if there exists a positive number c such that
c kvkV,2 ≤ kvkV,1 ≤ c−1 kvkV,2 ,
∀v ∈ V.
(A.2)
Remark A.4 (Finite dimension). If the vector space V has finite dimension, all the norms in V are equivalent. This result is false in infinitedimensional vector spaces.
⊓
⊔
Proposition A.5 (Compactness of unit ball). Let V be a normed vector
space and let B(0, 1) be the closed unit ball in V . Then, B(0, 1) is compact
(for the norm topology) if and only if V is finite-dimensional.
Proof. See Brezis [97, Thm. 6.5], Lax [321, §5.2].
⊓
⊔
Definition A.6 (Bounded linear maps). Let V and W be two normed
vector spaces. L(V ; W ) is the vector space of bounded linear maps from V to
W . The action of A ∈ L(V ; W ) on an element v ∈ V is denoted A(v) or,
more simply, Av. Maps in L(V ; W ) are often called operators.
Example A.7 (Continuous embedding). Let V and W be two normed
vector spaces. Assume that V ⊂ W and that there is c such that kvkW ≤
ckvkV for all v ∈ V . This property means that the embedding of V into W
is continuous. We say that V is continuously embedded into W and we write
V ֒→ W .
A.2 Banach spaces
Definition A.8 (Banach space). A vector space V equipped with a norm
k·kV such that every Cauchy sequence (with respect to the metric d(x, y) =
kx − ykV ) in V has a limit in V is called a Banach space.
A.2.1 Operators in Banach spaces
Proposition A.9 (Banach space). Let V be a normed vector space and let
W be a Banach space. Equip L(V ; W ) with the norm
kAkL(V ;W ) = sup
v∈V
kA(v)kW
,
kvkV
∀A ∈ L(V ; W ).
(A.3)
Then, L(V ; W ) is a Banach space.
Proof. See Rudin [399, p. 87], Yosida [483, p. 111].
⊓
⊔
Part XII. Appendices
775
Remark A.10 (Notation). In this book, we systematically abuse the noW
W
instead of supv∈V \{0} kA(v)k
⊓
⊔
tation by writing supv∈V kA(v)k
kvkV
kvkV .
The Uniform Boundedness Principle (or Banach–Steinhaus Theorem) is a
useful tool to study the limit of a sequence of operators in Banach spaces.
Theorem A.11 (Uniform Boundedness Principle). Let V and W be two
Banach spaces. Let {Ai }i∈I be a family (not necessarily countable) of operators
in L(V ; W ). Assume that
sup kAi vkW < ∞,
i∈I
∀v ∈ V.
(A.4)
Then, there is a constant C such that
kAi vkW ≤ CkvkV ,
∀v ∈ V,
∀i ∈ I.
(A.5)
Proof. See Brezis [97, p. 32], Lax [321, Chap. 10].
⊓
⊔
Corollary A.12 (Point-wise convergence). Let V and W be two Banach
spaces. Let (An )n∈N be a sequence of operators in L(V ; W ) such that, for all
v ∈ V , the sequence (An v)n∈N converges as n → ∞ to a limit in W denoted
Av (this means that the sequence (An )n∈N converges pointwise to A). Then,
the following holds:
(i) supn∈N kAn kL(V ;W ) < ∞;
(ii) A ∈ L(V ; W );
(iii) kAkL(V ;W ) ≤ lim inf n→∞ kAn kL(V ;W ) .
Proof. Statement (i) is just a consequence of Theorem A.11. Owing to (A.5),
we infer that kAn vkW ≤ CkvkV for all v ∈ V and all n ∈ N. Letting n → ∞,
we obtain that kAvkW ≤ CkvkV , and since A is obviously linear, we infer
that statement (ii) holds. Finally, statement (iii) results from the fact that
kAn vkW ≤ kAn kL(V ;W ) kvkV for all v ∈ V and all n ∈ N.
⊓
⊔
Remark A.13 (Uniform convergence on compact sets). Note that
Corollary A.12 does not claim that (An )n∈N converges to A in L(V ; W ),
i.e., uniformly on bounded sets. However, a standard argument shows that
(An )n∈N converges uniformly to A on compact sets. Indeed, let K ⊂ V be a
compact set. Let ǫ > 0. Set C := supn∈N kAn kL(V ;W ) ; this quantity is finite
owing to statement (i) in Corollary A.12. K being compact, we infer that
there is a finite set of points {xi }i∈I in K such that, for all v ∈ K, there is
i ∈ I such that kv − xi kV ≤ (6C)−1 ǫ. Owing to the pointwise convergence of
(An )n∈N to A, there is Ni such that, for all n ≥ Ni , kAn xi − Axi kW ≤ 31 ǫ.
Using the triangle inequality and statement (iii) above, we infer that
kAn v − AvkW ≤ kAn (v − xi )kW + kAn xi − Axi kW + kA(v − xi )kW ≤ ǫ,
for all v ∈ K and all n ≥ maxi∈I Ni .
⊓
⊔
776
Appendix A. Banach and Hilbert Spaces
Compact operators are encountered in various important situations, e.g.,
the Peetre–Tartar Lemma A.53 and the spectral theory developed in §A.5.1.
Definition A.14 (Compact operator). Let V and W be two Banach
spaces. T ∈ L(V ; W ) is called a compact operator if from every bounded
sequence (vn )n∈N in V , one can extract a subsequence (vnk )k∈N such that the
sequence (T vnk )k∈N converges in W ; equivalently, T maps the unit ball in V
into a relatively compact set in W .
Proposition A.15 (Composition with compact operator). Let W , X,
Y , Z be four Banach spaces and A ∈ L(Z; Y ), K ∈ L(Y ; X), B ∈ L(X; W ).
Assume that K is compact. Then B ◦ K ◦ A is compact.
Example A.16 (Compact injection). A classical example is the case
where V and W are two Banach spaces such that the injection of V into
W is compact. Then from every bounded sequence (vn )n∈N in V , one can
extract a subsequence that converges in W .
⊓
⊔
A.2.2 Duality
We start with real vector spaces and then discuss the extension to complex
vector spaces.
Definition A.17 (Dual space, Bounded linear forms). Let V be a
normed vector space over R. The dual space of V is defined to be L(V ; R)
and is denoted V ′ . An element A ∈ V ′ is called a bounded linear form. Its
action on an element v ∈ V is either denoted A(v) (or Av) or by means of
duality brackets in the form hA, viV ′ ,V for all v ∈ V .
Owing to Proposition A.9, V ′ is a Banach space when equipped with the
norm
|hA, viV ′ ,V |
|A(v)|
= sup
,
∀A ∈ V ′ .
(A.6)
kAkV ′ = sup
kvkV
v∈V
v∈V kvkV
Note that the absolute value can be omitted from the numerators since A is
linear and R-valued, and ±v can be considered in the supremum.
Theorem A.18 (Hahn–Banach). Let V be a normed vector space over R
and let W be a subspace of V . Let B ∈ W ′ = L(W ; R) be a bounded linear
B(w)
map with norm kBkW ′ = supw∈W kwk
. Then, there exists a bounded linear
V
′
form A ∈ V with the following properties:
(i) A is an extension of B, i.e., A(w) = B(w) for all w ∈ W .
(ii) kAkV ′ = kBkW ′ .
Proof. See Brezis [97, p. 3], Lax [321, Chap. 3], Rudin [399, p. 56], Yosida
[483, p. 102]. The above statement is a simplified version of the actual Hahn–
Banach Theorem.
⊓
⊔
Part XII. Appendices
777
Corollary A.19 (Dual characterization of norm). Let V be a normed
vector space over R. Then, the following holds:
kvkV =
sup
A(v) =
A∈V ′ ,kAkV ′ =1
sup
A∈V ′ ,kAkV ′ =1
hA, viV ′ V ,
(A.7)
for all v ∈ V , and the supremum is attained.
Proof. Assume v 6= 0 (the assertion is obvious for v = 0). We first observe that
supA∈V ′ ,kAkV ′ =1 A(v) ≤ kvkV . Let W = span(v) and let B ∈ W ′ be defined
as B(tv) = tkvkV for all t ∈ R. Owing to the Hahn–Banach Theorem, there
⊓
⊔
exists A ∈ V ′ such that kAkV ′ = kBkW ′ = 1 and A(v) = B(v) = kvkV .
Corollary A.20 (Characterization of density). Let V be a normed space
over R and let W be a subspace of V . Assume that any bounded linear form
in V ′ vanishing identically on W vanishes identically on V . Then, W = V .
⊓
⊔
Proof. See Brezis [97, p. 8], Rudin [399, Thm. 5.19].
Definition A.21 (Adjoint operator). Let V and W be two normed vector
spaces over R and let A ∈ L(V ; W ). The adjoint operator, or dual operator,
A∗ : W ′ → V ′ is defined by
hA∗ w′ , viV ′ ,V = hw′ , AviW ′ ,W ,
∀(v, w′ ) ∈ V × W ′ .
(A.8)
Definition A.22 (Double dual). The double dual of a Banach space V over
R is the dual of V ′ and is denoted V ′′ .
Proposition A.23 (Isometric embedding into V ′′ ). Let V be a Banach
space over R. Then, V ′′ is a Banach space, and the linear map JV : V → V ′′
defined by
hJV v, w′ iV ′′ ,V ′ = hw′ , viV ′ ,V ,
∀(v, w′ ) ∈ V × V ′ ,
(A.9)
is an isometry.
Proof. That V ′′ is a Banach space results from Proposition A.9. That JV is
an isometry results from
kJV vkV ′′ =
sup
w′ ∈V ′
kw′ k ′ =1
V
|hJV v, w′ iV ′′ ,V ′ | =
sup
w′ ∈V ′
kw′ k ′ =1
V
where the last equality results from Corollary A.19.
|hw′ , viV ′ ,V | = kvkV ,
⊓
⊔
Remark A.24 (Map JV ). Since the map JV is an isometry, it is injective.
As a result, V can be identified with the subspace JV (V ) ⊂ V ′′ . It may
happen that the map JV is not surjective. In this case, the space V is a
proper subspace of V ′′ . For instance, L∞ (D) = L1 (D)′ but L1 (D) ( L∞ (D)′
with strict inclusion; see §B.4 or Brezis [97, §4.3].
⊓
⊔
778
Appendix A. Banach and Hilbert Spaces
Definition A.25 (Reflexive Banach spaces). Let V be a Banach space
over R. V is said to be reflexive if JV is an isomorphism.
Let now V be a normed vector space over C. The notion of dual space of
V can be defined as in Definition A.17 by setting V ′ = L(V ; C). However, in
the context of weak formulations of PDEs with complex-valued functions, it is
more convenient to work with maps A : V → C that are antilinear ; this means
that A(v + w) = Av + Aw for all v, w ∈ V (as usual), but A(λv) = λv for all
λ ∈ C and all v ∈ V , where λ denotes the complex conjugate of λ (instead
of A(λv) = λv, in which case the map is linear). We denote by V ′ the vector
space of antilinear maps that are bounded with respect to the norm (A.6)
(note that we are now using the modulus in the numerators).
Our aim is to extend the result of Corollary A.19 to measure the norm of
the elements of V by the action of the elements of V ′ . To this purpose, it is
useful to consider V also as a vector space over R by restricting the scaling
λv to λ ∈ R and v ∈ V . The corresponding vector space is denoted VR to
distinguish it from V (thus, V and VR are the same sets, but equipped with
different structures). For instance, if V = Cm so that dim(V ) = m, then
dim(VR ) = 2m; a basis of V is the set {ek }1≤k≤m with ek,l = δkl (the Kronecker symbol) for all l ∈ {1: m}, while a basis of VR is the set {ek , iek }1≤k≤m
with i2 = −1. Another example is V = L2 (0, 2π; C) for which an Hilbertian
basis is the set {cos(nx), sin((n + 1)x)}n∈N , while an Hilbertian basis of VR is
the set {cos(nx), i cos(nx), sin((n + 1)x), i sin((n + 1)x)}n∈N .
Let VR′ be the dual space of VR , i.e., spanned by bounded R-linear maps
from V to R.
Lemma A.26 (Isometry for V ′ ). The map I : V ′ ∋ A 7→ I(A) ∈ VR′ such
that I(A)(v) = ℜ(A(v)) for all v ∈ V , is a bijective isometry.
Proof. The operator I(A) maps onto R and is linear since I(A)(tv) =
ℜ(A(tv)) = ℜ(tA(v)) = tℜ(A(v)) = tI(A)(v) for all t ∈ R and all v ∈ V .
Moreover, I(A) is bounded since
I(A)(v) = ℜ(A(v)) ≤ |A(v)| ≤ kAkV ′ kvkV ,
for all v ∈ V , so that kI(A)kVR′ ≤ kAkV ′ . Furthermore, the map I is injective
because ℜ(A(v)) = 0 for all v ∈ V implies ℜ(A(iv)) = 0, i.e., ℑ(A(v)) = 0 so
that A(v) = 0. Let us now prove that I is surjective. Let ψ ∈ VR′ and consider
the map A : V → C so that
A(v) = ψ(v) + iψ(iv),
∀v ∈ V.
(Recall that ψ is only R-linear.) By construction, I(A) = ψ, and the map
A : V → C is antilinear; indeed, for all λ ∈ C, writing λ = µ + iν with
µ, ν ∈ R, we infer that
Part XII. Appendices
779
A(λw) = ψ(µw + iνw) + iψ(iµw − νw)
= µψ(w) + νψ(iw) + iµψ(iw) − iνψ(w)
= µ (ψ(w) + iψ(iw)) − iν (ψ(w) + iψ(iw)) = λA(v),
for all w ∈ V , where we have used the R-linearity of ψ. Let us finally show
A(v)
∈ C.
that kAkV ′ ≤ kψkVR′ . Let v ∈ V be such that A(v) 6= 0 and set λ = |A(v)|
Then,
|A(v)| = λ−1 A(v) = A(λ−1 v) = ψ(λ−1 v) + iψ(iλ−1 v),
but since ψ takes values in R, we infer that ψ(iλ−1 v) = 0, so that |A(v)| =
ψ(λ−1 v). As a result,
|A(v)| ≤ kψkVR′ kλ−1 vkV = kψkVR′ kvkV ,
since |λ| = 1. This concludes the proof.
⊓
⊔
Corollary A.27 (Dual characterization of norm). Let V be a normed
vector space over C. Then, the following holds:
kvkV =
max
A∈V ′ ,kAkV ′ =1
ℜ(A(v)).
(A.10)
for all v ∈ V .
Proof. Combine the result of Corollary A.19 with Lemma A.26.
⊓
⊔
Remark A.28 (Use of modulus). Note that it is possible to replace (A.10)
by kvkV = maxA∈V ′ ,kAkV ′ =1 |A(v)| since it is always possible to multiply A
in the supremum by a unitary complex number so that A(v) is real and nonnegative.
⊓
⊔
Remark A.29 (Hahn–Banach). A version of the Hahn–Banach Theorem A.18 in complex vector spaces can be derived similarly to the above
construction; see Lax [321, p. 27].
⊓
⊔
The rest of the material is adapted straightforwardly. The adjoint of an
operator A ∈ L(V ; W ) is still defined by (A.8), and one can verify that it
maps (linearly) bounded antilinear maps in W ′ to bounded antilinear maps
in V ′ . Moreover, the bidual is defined by considering bounded antilinear forms
on V ′ , and the linear isometry extending that from Proposition A.23 is such
that hJV v, w′ iV ′′ ,V ′ = hw′ , viV ′ ,V .
A.2.3 Interpolation between Banach spaces
Interpolating between Banach spaces is a useful tool to bridge between known
results so as to derive new results that could difficult to obtain directly. An
important application is the derivation of interpolation error estimates in
fractional-order Sobolev spaces. There are many interpolation methods; see,
780
Appendix A. Banach and Hilbert Spaces
e.g., Bergh and Löfström [47], Tartar [443] and references therein. For simplicity, we focus here on the real interpolation K-method; see [47, §3.1] and
[443, Chap. 22].
Let V0 and V1 be two normed vector spaces, continuously embedded into
a common topological vector space V. Then, V0 + V1 is a normed vector space
with the (canonical) norm kvkV0 +V1 = inf v=v0 +v1 (kv0 kV0 +kv1 kV1 ). Moreover,
if V0 and V1 are Banach spaces, then V0 + V1 is also a Banach space; see [47,
Lem. 2.3.1]. For all v ∈ V0 + V1 and all t > 0, define
K(t, v) =
inf
v=v0 +v1
(kv0 kV0 + tkv1 kV1 ).
(A.11)
For all t > 0, v 7→ K(t, v) defines a norm on V0 +V1 equivalent to the canonical
norm. One can also verify that t 7→ K(t, v) is nondecreasing and concave (and
therefore continuous) and that t 7→ 1t K(t, v) is increasing.
Definition A.30 (Interpolated space). Let θ ∈ (0, 1) and let p ∈ [1, ∞].
The interpolated space [V0 , V1 ]θ,p is defined to be
[V0 , V1 ]θ,p = {v ∈ V0 + V1 | kt−θ K(t, v)kLp (R+ ; dt ) < ∞},
t
(A.12)
p1
R∞
where kϕkLp (R+ ; dt ) = 0 |ϕ(t)|p dt
for p ∈ [1, ∞) and kϕkL∞ (R+ ; dt ) =
t
t
t
sup0<t<∞ |ϕ(t)|. This space is equipped with the norm
kvk[V0 ,V1 ]θ,p = kt−θ K(t, v)kLp (R+ ; dt ) .
t
If V0 and V1 are Banach spaces, so is [V0 , V1 ]θ,p .
Remark A.31 (Value for θ). Since K(t, v) ≥ min(1, t)kvkV0 +V1 , the space
[V0 , V1 ]θ,p reduces to {0} if t−θ min(1, t) 6∈ Lp (R+ ; dt
t ). In particular, [V0 , V1 ]θ,p
is trivial if θ ∈ {0, 1} and p < ∞.
⊓
⊔
Remark A.32 (Gagliardo set). The map t 7→ K(t, v) has a simple geometric interpretation. Introducing the Gagliardo set G(v) = {(x0 , x1 ) ∈ R2 | v =
v0 +v1 with kv0 kV0 ≤ x0 and kv1 kV1 ≤ x1 }, one can verify that G(v) is convex
and that K(t, v) = inf v∈∂G(v) (x0 + tx1 ), so that the map t 7→ K(t, v) is one
way to explore the boundary of G(v); see [47, p. 39].
⊓
⊔
Remark A.33 (Intersection). The vector space V0 ∩ V1 can be equipped
with the (canonical) norm kvkV0 ∩V1 = max(kvkV0 , kvkV1 ). For all v ∈ V0 ∩ V1 ,
one can verify that K(t, v) ≤ min(1, t)kvkV0 ∩V1 , whence we infer the continuous embedding V0 ∩ V1 ֒→ [V0 , V1 ]θ,p for all θ ∈ (0, 1) and p ∈ [1, ∞]. As a
result, if V0 ⊂ V1 , then V0 ֒→ [V0 , V1 ]θ,p .
⊓
⊔
Lemma A.34 (Continuous embedding). Let θ ∈ (0, 1) and p, q ∈ [1, ∞]
with p ≤ q. Then, [V0 , V1 ]θ,p ֒→ [V0 , V1 ]θ,q .
Part XII. Appendices
781
Theorem A.35 (Riesz–Thorin, interpolation of operators). Let A :
V0 + V1 → W0 + W1 be a linear operator that maps V0 and V1 boundedly to
W0 and W1 . Then, for all θ ∈ (0, 1) and all p ∈ [1, ∞], A maps [V0 , V1 ]θ,p
boundedly to [W0 , W1 ]θ,p . Moreover,
1−θ
kAkL([V0 ,V1 ]θ,p ;[W0 ,W1 ]θ,p ) ≤ kAkL(V
kAkθL(V1 ;W1 ) .
0 ;W0 )
(A.13)
⊓
⊔
Proof. See [443, Lem. 22.3].
Theorem A.36 (Lions–Peetre, reiteration). Let θ0 , θ1 ∈ [0, 1] with
θ0 6= θ1 . Assume that [V0 , V1 ]θ0 ,1 ֒→ W0 ֒→ [V0 , V1 ]θ0 ,∞ and [V0 , V1 ]θ1 ,1 ֒→
W1 ֒→ [V0 , V1 ]θ1 ,∞ . Then, for all θ ∈ (0, 1) and all p ∈ [1, ∞], [W0 , W1 ]θ,p =
[V0 , V1 ]η,p with equivalent norms where η = (1 − θ)θ0 + θθ1 .
⊓
⊔
Proof. See Tartar [443, Thm. 26.2].
Theorem A.37 (Lions–Peetre, extension). Let V0 , V1 , F be three Banach
spaces. Let A ∈ L(V0 ∩V1 ; F ), then A extends into a linear continuous mapping
from [V0 , V1 ]θ,1;J to F if and only if
∃c < ∞, ∀v ∈ V0 ∩ V1
kAvkF ≤ ckvkV1−θ
kvkθV1 .
0
Proof. See [443, Lem. 25.3].
(A.14)
⊓
⊔
Theorem A.38 (Interpolation of dual spaces). Let θ ∈ (0, 1) and p ∈
p
(with the convention
[1, ∞). Then, [V0 , V1 ]′θ,p = [V1′ , V0′ ]1−θ,p′ where p′ = p−1
′
that p = ∞ if p = 1).
Proof. See Bergh and Löfström [47].page!
⊓
⊔
A.3 Hilbert spaces
We start with real vector spaces and then briefly discuss the extension to
complex vector spaces.
Definition A.39 (Inner product). Let V be a vector space over R. An inner
product (or scalar product) on V is a map
(·, ·)V : V × V ∋ (v, w) 7−→ (v, w)V ∈ R,
(A.15)
satisfying the following three properties:
(i) Bilinearity: (v, w)V is a linear function of w ∈ V for fixed v ∈ V , and it
is a linear function of v ∈ V for fixed w ∈ V .
(ii) Symmetry: (v, w)V = (w, v)V for all v, w ∈ V .
(iii) Positive definiteness: (v, v)V ≥ 0 for all v ∈ V and (v, v)V = 0 ⇐⇒
v = 0.
(JLG) I changed
θ0 , θ1 ∈ (0, 1) to
θ0 , θ1 ∈ [0, 1] since
this is what we need
in general.
782
Appendix A. Banach and Hilbert Spaces
Proposition A.40 (Cauchy–Schwarz). Let (·, ·)V be an inner product on
the real vector space V . By setting
1
kvkV = (v, v)V2 ,
∀v ∈ V,
(A.16)
one defines a norm on V . Moreover, the Cauchy–Schwarz1 inequality holds:
|(v, w)V | ≤ kvkV kwkV ,
∀v, w ∈ V.
(A.17)
Remark A.41 (Equality). The Cauchy–Schwarz inequality can be seen as a
v
w 2
consequence of the identity kvkV kwkV −(v, w)V = kvkV 2kwkV kvk
,
− kwk
V
V
V
valid for all non-zero v, w in V . This identity shows that equality holds in
(A.17) if and only if v and w are collinear.
⊓
⊔
Proposition A.42 (Arithmetic-geometric inequality). Let x1 , . . . , xn be
non-negative numbers. Then,
1
1
n (x1
(x1 x2 . . . xn ) n ≤
+ . . . + xn ).
Proof. Use the convexity of the function x 7→ ex .
(A.18)
⊓
⊔
This inequality is frequently used in conjunction with the Cauchy–Schwarz
inequality. In particular, it implies that
|(v, w)V | ≤ γ2 kvk2V +
2
1
2γ kwkV
,
∀γ > 0, ∀v, w ∈ V.
(A.19)
Definition A.43 (Hilbert spaces). A Hilbert space V is an inner product
space over R that is complete with respect to the induced norm (and is, therefore, a Banach space). The inner product is denoted (·, ·)V and the induced
norm k·kV . A Hilbert space is said to be separable if it admits a countable and
dense subset.
Theorem A.44 (Riesz–Fréchet). Let V be a Hilbert space over R. For each
v ′ ∈ V ′ , there exists a unique u ∈ V such that
hv ′ , wiV ′ ,V = (u, w)V ,
∀w ∈ V.
(A.20)
Moreover, the map v ′ ∈ V ′ 7→ u ∈ V is an isometric isomorphism.
Proof. See Brezis [97, Thm. 5.5], Lax [321, p. 56], Yosida [483, p. 90].
⊓
⊔
An important consequence of the Riesz–Fréchet Theorem is the following:
Proposition A.45 (Reflexivity). Hilbert spaces are reflexive.
Proof. Let V be a Hilbert space. The Riesz–Fréchet Theorem implies that V
can be identified with V ′ ; similarly, V ′ can be identified with V ′′ .
⊓
⊔
1
Augustin-Louis Cauchy (1789–1857) and Herman Schwarz (1843–1921)
Part XII. Appendices
783
Proposition A.46 (Orthogonal projection). Let V be a Hilbert space over
R. Let U be a non-empty, closed, and convex subset of V . Let f ∈ V .
(i) There is a unique u ∈ U such that kf − ukV = minv∈U kf − vkV .
(ii) This unique minimizer is characterized by the Euler–Lagrange condition
(f − u, v − u)V ≤ 0 for all v ∈ U .
(iii) In the case where U is a subspace of V , the unique minimizer is characterized by the condition (f − u, v)V = 0 for all v ∈ U , and the map
ΠU : V ∋ f 7→ u ∈ U is linear with kΠU kL(V ;U ) = 1 (unless U = {0} so
that kΠU kL(V ;U ) = 0).
Proof. See Exercise 17.3.
⊓
⊔
Let now V be a vector space over C. Then, an inner product on V is a
map (·, ·)V : V × V ∋ (v, w) 7−→ (v, w)V ∈ C satisfying the following three
properties:
(i) Sesquilinearity: (v, w)V is an antilinear function of w ∈ V for fixed v ∈ V ,
and it is a linear function of v ∈ V for fixed w ∈ V .
(ii) Hermitian symmetry: (v, w)V = (w, v)V for all v, w ∈ V .
(iii) Positive definiteness: (v, v)V ≥ 0 for all v ∈ V and (v, v)V = 0 ⇐⇒ v =
0 (note that (v, v)V is always real owing to Hermitian symmetry).
The extension of the above results from real to complex Hilbert spaces is as
1/2
follows. The map v 7→ (v, v)V still defines a norm on V , and the Cauchy–
Schwarz inequality still takes the form (A.17) (with the modulus on the lefthand side). The Riesz–Fréchet Theorem A.44 still states that, for all v ′ ∈ V ′
(recall that v ′ is antilinear by convention), there exists a unique u ∈ V such
that hv ′ , wiV ′ ,V = (u, w)V for all w ∈ V , and the map v ′ ∈ V ′ 7→ u ∈ V
is an isometric (linear) isomorphism. Finally, the Euler–Lagrange condition
from Proposition A.46 now becomes ℜ(f − u, v − u)V ≤ 0 for all v ∈ U . The
proof of these extensions hinges on the fact that, if V is a complex Hilbert
space equipped with the inner product (·, ·)V , then VR equipped with the inner
product ℜ(·, ·)V is a real Hilbert space (recall that V and VR are the same
sets, equipped with different structures). Let us prove for instance the Riesz–
Fréchet Theorem. Let v ′ ∈ V ′ . Using the bijective isometry from Lemma A.26,
we consider I(v ′ ) ∈ VR′ . Owing to Theorem A.44 on VR equipped with ℜ(·, ·)V ,
there is a unique u ∈ VR such that ℜ(u, w)V = I(v ′ )(w) = ℜhv ′ , wiV ′ ,V for all
w ∈ VR . Considering w and iw yields (u, w) = hv ′ , wiV ′ ,V for all w ∈ V .
A.4 Bijective Banach operators
Let V and W be two Banach spaces. Maps in L(V ; W ) are called (linear)
Banach operators. This section presents classical results to characterize bijective linear Banach operators, see Aubin [24], Brezis [97], Yosida [483]. Some
784
Appendix A. Banach and Hilbert Spaces
of the material presented herein is adapted from Azerad [26], Guermond and
Quartapelle [264]. For simplicity, we implicitly assume that V and W are real
Banach spaces, and we briefly indicate relevant changes in the complex case.
A.4.1 Fundamental results
For A ∈ L(V ; W ), we denote by ker(A) its kernel and by im(A) its range. The
operator A being bounded, ker(A) is closed in V . Hence, the quotient of V
by ker(A), V /ker(A), can be defined. This space is composed of equivalence
classes v̌ such that v and w are in the same class v̌ if and only if v−w ∈ ker(A).
Theorem A.47 (Quotient space). The space V /ker(A) is a Banach space
when equipped with the norm kv̌k = inf v∈v̌ kvkV . Moreover, defining Ǎ :
V /ker(A) → im(A) by Ǎv̌ = Av for all v in v̌, Ǎ is an isomorphism.
Proof. See Brezis [97, §11.2], Yosida [483, p. 60].
⊓
⊔
For subspaces M ⊂ V and N ⊂ V ′ , we introduce the so-called annihilators
of M and N which are defined as follows:
M ⊥ = {v ′ ∈ V ′ | ∀m ∈ M, hv ′ , miV ′ ,V = 0},
N
⊥
′
′
= {v ∈ V | ∀n ∈ N, hn , viV ′ ,V = 0}.
(A.21)
(A.22)
A characterization of ker(A) and im(A) is given by the following:
Lemma A.48 (Kernel and range). For A in L(V ; W ), the following properties hold:
(i)
(ii)
(iii)
(iv)
ker(A) = (im(A∗ ))⊥ .
ker(A∗ ) = (im(A))⊥ .
im(A) = (ker(A∗ ))⊥ .
im(A∗ ) ⊂ (ker(A))⊥ .
Proof. See Brezis [97, Cor. 2.18], Yosida [483, p. 202-209].
⊓
⊔
Showing that the range of an operator is closed is a crucial step towards
proving that this operator is surjective. This is the purpose of the following
fundamental theorem:
Theorem A.49 (Banach or Closed Range). Let A ∈ L(V ; W ). The following statements are equivalent:
(i)
(ii)
(iii)
(iv)
im(A) is closed.
im(A∗ ) is closed.
im(A) = (ker(A∗ ))⊥ .
im(A∗ ) = (ker(A))⊥ .
Proof. See Brezis [97, Thm. 2.19], Yosida [483, p. 205].
⊓
⊔
Part XII. Appendices
785
We now put in place the second keystone of the edifice:
Theorem A.50 (Open Mapping). If A ∈ L(V ; W ) is surjective and U is
an open set in V , then A(U ) is open in W .
Proof. See Brezis [97, Thm. 2.6], Lax [321, p. 168], Rudin [399, p. 47], Yosida
[483, p. 75].
⊓
⊔
Theorem A.50, also due to Banach, has far-reaching consequences. In particular, we deduce the following:
Lemma A.51 (Characterization of closed range). Let A ∈ L(V ; W ).
The following statements are equivalent:
(i) im(A) is closed.
(ii) There exists α > 0 such that
∀w ∈ im(A), ∃vw ∈ V,
Avw = w
and
αkvw kV ≤ kwkW .
(A.23)
Proof. (i) ⇒ (ii). Since im(A) is closed in W , im(A) is a Banach space. Applying the Open Mapping Theorem to A : V → im(A) and U = BV (0, 1)
(the open unit ball in V ) yields that A(BV (0, 1)) is open in im(A). Since
0 ∈ A(BV (0, 1)), there is γ > 0 such that BW (0, γ) ⊂ A(BV (0, 1)). Let
w
∈ BW (0, α), there is z ∈ BV (0, 1) such that
w ∈ im(A). Since γ2 kwk
W
w
W
Az = γ2 kwk
. Setting v = 2kwk
z leads Av = w and γ2 kvkV ≤ kwkW .
γ
W
(ii) ⇒ (i). Let (wn )n∈N be a sequence in im(A) that converges to some w ∈ W .
Using (A.23), we infer that there exists a sequence (vn )n∈N in V such that
Avn = wn and αkvn kV ≤ kwn kW . Then, (vn )n∈N is a Cauchy sequence in V .
Since V is a Banach space, (vn )n∈N converges to a certain v ∈ V . Owing to
the boundedness of A, (Avn )n∈N converges to Av. Hence, w = Av ∈ im(A),
proving statement (i).
⊓
⊔
Remark A.52 (Bounded inverse). A first consequence of Lemma A.51
is that if A ∈ L(V ; W ) is bijective, then its inverse is necessarily bounded.
Indeed, the fact that A is bijective implies that A is injective and im(A)
is closed. Lemma A.51 implies that there is α > 0 such that kA−1 wkV ≤
1
−1
is bounded.
⊓
⊔
α kwkW , i.e., A
Let us finally give a sufficient condition for the image of an injective operator to be closed.
Lemma A.53 (Peetre–Tartar). Let X, Y , Z be three Banach spaces. Let
A ∈ L(X; Y ) be an injective operator and let T ∈ L(X; Z) be a compact
operator. If there is c > 0 such that ckxkX ≤ kAxkY + kT xkZ , then im(A) is
closed; equivalently, there is α > 0 such that
∀x ∈ X,
αkxkX ≤ kAxkY .
(A.24)
786
Appendix A. Banach and Hilbert Spaces
Proof. By contradiction. Assume that there is a sequence (xn )n∈N of X such
that kxn kX = 1 and kAxn kY converges to zero when n goes to infinity. Since
T is compact and the sequence (xn )n∈N is bounded, there is a subsequence
(xnk )k∈N such that (T xnk )k∈N is a Cauchy sequence in Z. Owing to the inequality
αkxnk − xmk kX ≤ kAxnk − Axmk kY + kT xnk − T xmk kZ ,
(xnk )k∈N is a Cauchy sequence in X. Let x be its limit. Clearly, kxkX = 1.
The boundedness of A implies Axnk → Ax and Ax = 0 since Axnk → 0. Since
A is injective x = 0, which contradicts the fact that kxkX = 1.
⊓
⊔
A.4.2 Characterization of surjectivity
As a consequence of the Closed Range Theorem and of the Open Mapping
Theorem, we deduce two lemmas characterizing surjective operators.
Lemma A.54 (Surjectivity of A∗ ). Let A ∈ L(V ; W ). The following statements are equivalent:
(i) A∗ : W ′ → V ′ is surjective.
(ii) A : V → W is injective and im(A) is closed in W .
(iii) There exists α > 0 such that
∀v ∈ V,
kAvkW ≥ αkvkV ,
(A.25)
or, equivalently, there exists α > 0 such that
inf sup
v∈V w′ ∈W ′
hw′ , AviW ′ ,W
≥ α.
kw′ kW ′ kvkV
(A.26)
In the complex case, real parts of duality brackets are considered.
Proof. (i) ⇒ (iii). The Open Mapping Theorem implies that, for all v ′ ∈ V ′ ,
there is wv′ ′ ∈ W ′ such that A∗ wv′ ′ = v ′ and kwv′ ′ kW ′ ≤ α−1 kv ′ kV ′ . Let now
v ∈ V . Then,
′
hA∗ wv′ ′ , viV ′ ,V
hv ′ , viV ′ ,V
−1 hwv ′ , AviW ′ ,W
=
≤
α
kv ′ kV ′
kv ′ kV ′
kwv′ ′ kW ′
hw′ , AviW ′ ,W
.
≤ α−1 sup
kw′ kW ′
w′ ∈W ′
Taking the supremum in v ′ ∈ V ′ yields (A.26) since
kvkV = sup
v ′ ∈V ′
hv ′ , viV ′ ,V
hw′ , AviW ′ ,W
≤ α−1 sup
.
′
kv kV ′
kw′ kW ′
w′ ∈W ′
(iii) ⇒ (ii). The bound (A.25) implies that A is injective. To prove that im(A)
is closed, consider a sequence (vn )n∈N such that (Avn )n∈N is a Cauchy sequence in W . Then, (A.25) implies that (vn )n∈N is a Cauchy sequence in V .
Part XII. Appendices
787
Let v be its limit. A being bounded implies that Avn → Av; hence, im(A) is
closed.
(ii) ⇒ (i). Since im(A) is closed, we use Theorem A.49(iv) together with the
injectivity of A to infer that im(A∗ ) = {0}⊥ = V ′ .
⊓
⊔
Lemma A.55 (Surjectivity of A). Let A ∈ L(V ; W ). The following statements are equivalent:
(i) A : V → W is surjective.
(ii) A∗ : W ′ → V ′ is injective et im(A∗ ) is closed in V ′ .
(iii) There exists α > 0 such that
∀w′ ∈ W ′ ,
kA∗ w′ kV ′ ≥ αkw′ kW ′ ,
(A.27)
or, equivalently, there exists α > 0 such that
inf sup
w′ ∈W ′
v∈V
hA∗ w′ , viV ′ ,V
≥ α.
kw′ kW ′ kvkV
(A.28)
In the complex case, real parts of duality brackets are considered.
⊓
⊔
Proof. Similar to that of Lemma A.54.
Remark A.56 (Lions’ Theorem). The statement (i) ⇔ (iii) in Lemma A.55
is sometimes referred to as Lions’ Theorem. Establishing the a priori estimate
(A.28) is a necessary and sufficient condition to prove that the problem Au = f
has at least one solution u in V for all f in W .
⊓
⊔
One easily verifies (see Lemma A.57) that (A.23) implies the inf-sup condition (A.28). In practice, however, it is often easier to check condition (A.28)
than to prove that for all w ∈ im(A), there exists an inverse image vw satisfying (A.23). At this point, the natural question that arises is to determine
whether the constant α in (A.28) is the same as that in (A.23). The answer to
this question is the purpose of the next lemma which is due to Azerad [26, 27].
Lemma A.57 (Inf-sup condition). Let V and W be two Banach spaces
and let A ∈ L(V ; W ) be a surjective operator. Let α > 0. The property
∀w ∈ W, ∃vw ∈ V,
Avw = w
implies
inf sup
w′ ∈W ′
v∈V
and
αkvw kV ≤ kwkW ,
hA∗ w′ , viV ′ ,V
≥ α.
kw′ kW ′ kvkV
(A.29)
(A.30)
The converse is true if V is reflexive. In the complex case, it is the real part
of the duality bracket that is considered.
788
Appendix A. Banach and Hilbert Spaces
Proof. (1) The implication. By definition of the norm in W ′ ,
∀w′ ∈ W ′ ,
kw′ kW ′ =
sup hw′ , wiW ′ ,W .
w∈W
kwkW =1
For all w in W , there is vw ∈ V such that Avw = w and αkvw kV ≤ kwkW .
Let w′ in W ′ . Therefore,
hw′ , wiW ′ ,W = hw′ , Avw iW ′ ,W = hA∗ w′ , vw iV ′ ,V ≤
Hence,
kw′ kW ′ =
sup hw′ , wiW ′ ,W ≤
w∈W
kwkW =1
∗ ′
1
′
α kA w kV kwkW .
∗ ′
1
′
α kA w kV .
The desired inequality follows from the definition of the norm in V ′ .
(2) Let us prove the converse statement under the assumption that V is reflexive. The inf-sup inequality being equivalent to kA∗ w′ kV ′ ≥ αkw′ kW ′ for
all w′ ∈ W ′ , A∗ is injective. Let v ′ ∈ im(A∗ ) and define z ′ (v ′ ) ∈ W ′ such that
A∗ (z ′ (v ′ )) = v ′ . Note that z ′ (v ′ ) is unique since A∗ is injective; this in turn
implies that z ′ (·) : im(A∗ ) ⊂ V ′ → W ′ is a linear mapping. (Note also that
z ′ is injective: assume the 0 = z ′ (v ′ ), then 0 = A∗ (z ′ (v ′ )) = v ′ . The mapping is also surjective: let w′ ∈ W ′ , then A∗ (z ′ (A∗ w′ )) = A∗ w′ , which implies
z ′ (A∗ w′ ) = w′ since A∗ is surjective.) In conclusion z ′ (·) : im(A∗ ) ⊂ V ′ → W ′
is an isomorphism. Let w ∈ W and let us construct an inverse image for w,
say vw , satisfying (A.23). We first define the linear form φw : im(A∗ ) → R by
∀v ′ ∈ im(A∗ ),
φw (v ′ ) = hz ′ (v ′ ), wiW ′ ,W ,
i.e., φw (A∗ w′ ) = hw′ , wiW ′ ,W for all w′ ∈ W ′ . Hence,
|φw (v ′ )| ≤ kz ′ (v ′ )kW ′ kwkW ≤ α1 kA∗ z ′ (v ′ )kV ′ kwkW ≤
′
1
′
α kv kV kwkW .
This means that φw is bounded on im(A∗ ) equipped with the norm of V ′ .
Owing to the Hahn–Banach Theorem, φw can be extended to V ′ with the
same norm. Let φew ∈ V ′′ be the extension in question with kφew kV ′′ ≤ α1 kwkW .
Since V is assumed to be reflexive, there is vw ∈ V such that JV (vw ) = φew .
As a result,
∀w′ ∈ W ′ ,
hw′ , Avw iW ′ ,W = hA∗ w′ , vw iV ′ ,V = hJV (vw ), A∗ w′ iV ′′ ,V ′
= hφew , A∗ w′ iV ′′ ,V ′ = φw (A∗ w′ )
= hz ′ (A∗ w′ ), wiW ′ ,W = hw′ , wiW ′ ,W ,
showing that Avw = w. Hence, vw is an inverse image of w and
kvw kV = kJV (vw )kV ′′ = kφew kV ′′ ≤
1
α kwkW .
⊓
⊔
Remark A.58 (Linearity). Note that the dependence of vw with respect
to w in Lemma A.57 is a priori nonlinear. Linearity can be obtained when
the setting is Hilbertian by using the zero extension of φw on the orthogonal
complement of im(A∗ ) instead of invoking the Hahn–Banach Theorem.
⊓
⊔
Part XII. Appendices
789
A.4.3 Characterization of bijectivity
The following theorem provides the theoretical foundation of the BNB Theorem of §17.3.
Theorem A.59 (Bijectivity of A). Let A ∈ L(V ; W ). The following statements are equivalent:
(i) A : V → W is bijective.
(ii) A is injective, im(A) is closed, and A∗ : W ′ → V ′ is injective.
(iii) A∗ is injective and there exists α > 0 such that
kAvkW ≥ αkvkV ,
∀v ∈ V,
(A.31)
or, equivalently, A∗ is injective and
inf sup
v∈V w′ ∈W ′
hw′ , AviW ′ ,W
=: α > 0.
kw′ kW ′ kvkV
(A.32)
In the complex case, real parts of duality brackets are considered.
Proof. (1) Statements (ii) and (iii) are equivalent since (A.31) is equivalent
to A injective and im(A) closed owing to Lemma A.54.
(2) Let us first prove that (i) implies (ii). Since A is surjective, ker(A∗ ) =
im(A)⊥ = {0}, i.e., A∗ is injective. Since im(A) = W is closed and A is
injective, this yields (ii). Finally, to prove that (ii) implies (i), we only need
to prove that (ii) implies the surjectivity of A. The injectivity of A∗ implies
im(A) = (ker(A∗ ))⊥ = W . Since im(A) is closed, im(A) = W , i.e., A is
surjective.
⊓
⊔
Remark A.60 (Bijectivity of A∗ ). The bijectivity of A ∈ L(V ; W ) is
equivalent to that of A∗ ∈ L(W ′ ; V ′ ). Indeed, statement (ii) in Theorem A.59
is equivalent to A∗ injective and A∗ surjective owing to the equivalence of
statements (i) and (ii) from Lemma A.54.
⊓
⊔
Corollary A.61 (Inf-sup condition). Let A ∈ L(V ; W ) be a bijective operator. Assume that V is reflexive. Then,
inf sup
v∈V w′ ∈W ′
hw′ , AviW ′ ,W
hw′ , AviW ′ ,W
= ′inf ′ sup
.
′
w ∈W v∈V kw ′ kW ′ kvkV
kw kW ′ kvkV
(A.33)
In the complex case, real parts of duality brackets are considered.
Proof. The left-hand side, l, and the right-hand side, r, of (A.33) are two
positive finite numbers, since A is a bijective bounded operator. The lefthand side being equal to l means that l is the largest number such that
kAvkW ≥ l kvkV for all v in V . Let w′ ∈ W ′ and w ∈ W . Since A is surjective,
there is vw ∈ V so that Avw = w and the previous statement regarding l
implies that l kvw kV ≤ kwkW . This in turn implies that
790
Appendix A. Banach and Hilbert Spaces
hw′ , Avw iW ′ ,W
hA∗ w′ , vw iV ′ ,V
hw′ , wiW ′ ,W
= sup
= sup
kwkW
kwkW
kwkW
w∈W
w∈W
w∈W
1
kv
k
w V
≤ kA∗ w′ kV ′ ,
≤ kA∗ w′ kV ′ sup
l
w∈W kwkW
kw′ kW ′ = sup
which implies l ≤ r. That r ≤ l is proved similarly by working with W ′ in lieu
of V , V ′ in lieu of W and A∗ in lieu of A. The above reasoning leads to
inf
′
sup
w ∈W ′ v ′′ ∈V ′′
hv ′′ , A∗ w′ iV ′′ ,V ′
hv ′′ , A∗ w′ iV ′′ ,V ′
≤
inf
sup
,
v ′′ ∈V ′′ w′ ∈W ′ kv ′′ kV ′′ kw ′ kW ′
kv ′′ kV ′′ kw′ kW ′
⊓
⊔
and we conclude using the reflexivity of V .
Remark A.62 (Counter-example). Note that (A.33) may not hold if A 6=
0 is not bijective. For instance if A : (x1 , x2 , x3 . . .) 7−→ (0, x1 , x2 , x3 , . . .) is
the right shift operator in ℓ2 , then A∗ : (x1 , x2 , x3 . . .) 7−→ (x2 , x3 , x4 , . . .) is
the left shift operator. It can be verified that A is injective but not surjective
whereas A∗ is injective but not surjective. It can also be shown that l = 1 and
r = 0.
⊓
⊔
A.4.4 Coercive operators
We now focus on the smaller class of coercive operators.
Definition A.63 (Coercive operator). Let V be a Banach space over R.
A ∈ L(V ; V ′ ) is said to be a coercive operator if there exist a number α > 0
and ξ = ±1 such that
ξhAv, viV ′ ,V ≥ αkvk2V ,
∀v ∈ V.
(A.34)
In the complex case, A ∈ L(V ; V ′ ) is said to be a coercive operator if there
exist a real number α > 0 and a complex number ξ with |ξ| = 1 such that
ℜ (ξhAv, viV ′ ,V ) ≥ αkvk2V ,
∀v ∈ V.
(A.35)
The following proposition shows that the notion of coercivity is relevant
only in Hilbert spaces:
Proposition A.64 (Hilbert structure). Let V be a Banach space. V can
be equipped with a Hilbert structure with the same topology if and only if there
is a coercive operator in L(V ; V ′ ).
Proof. See Exercise 17.5.
⊓
⊔
Corollary A.65 (Sufficient condition). Coercivity is a sufficient condition
for an operator A ∈ L(V ; V ′ ) to be bijective.
Proof. Corollary A.65 is the Lax–Milgram Lemma; see §17.2.
⊓
⊔
Part XII. Appendices
791
We now introduce the class of self-adjoint operators.
Definition A.66 (Self-adjoint operator). Let V be a reflexive Banach
space, so that V and V ′′ are identified. The operator A ∈ L(V ; V ′ ) is said
to be self-adjoint if A∗ = A in the real case and if A∗ = A in the complex
case.
Self-adjoint bijective operators are characterized as follows:
Corollary A.67 (Self-adjoint bijective operator). Let V be a reflexive
Banach space and let A ∈ L(V ; V ′ ) be a self-adjoint operator. Then, A is
bijective if and only if there is a number α > 0 such that
kAvkV ′ ≥ αkvkV ,
∀v ∈ V.
(A.36)
Proof. Owing to Theorem A.59, the bijectivity of A implies that A satisfies
inequality (A.36). Conversely, inequality (A.36) means that A is injective. It
follows that A∗ is injective since A∗ = A (or A∗ = A) by hypothesis. The
conclusion is then a consequence of Theorem A.59(iii).
⊓
⊔
We finally introduce the concept of monotonicity.
Definition A.68 (Monotone operator). Let V be a Banach space over R.
The operator A ∈ L(V ; V ′ ) is said to be monotone if
hAv, viV ′ ,V ≥ 0,
∀v ∈ V.
In the complex case, the condition becomes ℜ(hAv, viV ′ ,V ) ≥ 0 for all v ∈ V .
Corollary A.69 (Equivalent condition). Let V be a reflexive Banach
space and let A ∈ L(V ; V ′ ) be a monotone self-adjoint operator. Then, A
is bijective if and only if A is coercive (with ξ = 1).
Proof. See Exercise 17.6.
⊓
⊔
The rest of this
chapter still to be
checked
A.5 Spectral theory
We briefly recall in this section some essential facts regarding the spectral
theory of linear operators. The material is classical and can be found in Brezis
[97, Chap. 6], Chatelin [131, p. 95-120], Dunford and Schwartz [201, Part I,
pp. 577-580], Lax [321, Chap. 21&32].
Definition A.70 (Resolvent, spectrum). Let L be a complex Banach space
and let T ∈ L(L; L). The resolvent set, ρ(T ), and the spectrum of T , σ(T ),
are the sets in C such that
792
Appendix A. Banach and Hilbert Spaces
ρ(T ) = {z ∈ C; (zI − T )−1 ∈ L(L; L)},
σ(T ) = C\ρ(T ).
(A.37)
(A.38)
µ ∈ C is called eigenvalue of T if ker(µI − T ) 6= {0} and ker(µI − T ) is the
associated eigenspace. The set of eigenvalues is denoted ε(T ).
Theorem A.71 (Gelfand). Let T ∈ L(L; L), then
(i) ρ(T ) and σ(T ) are nonempty.
(ii) σ(T ) is compact in C.
1
n
where |σ(T )| := maxλ∈σ(T ) |λ| is called the
(iii) |σ(T )| = limn→∞ kT n kL(L;L)
spectral radius of T .
A.5.1 Compact operators
We start with the following important results regarding compact operators.
Theorem A.72 (Schauder). A bounded linear operator between Banach
spaces is compact if and only if its adjoint is.
Theorem A.73 (Fredholm alternative). Let T ∈ L(L; L) be a compact
operator and µ ∈ C \ {0}. µI − T is injective if and only if µI − T is surjective.
The Fredholm alternative is often reformulated in the following equivalent
way: Either µI − T is bijective or ker(µI − T ) 6= 0. The key result for compact
operators is the following (see, e.g., [321, p. 238]).
Theorem A.74 (Spectrum). Let T ∈ L(L; L) be a compact operator, then
(i) σ(T ) is a countable set with no accumulation point other than zero.
(ii) Each nonzero member of σ(T ) is an isolated eigenvalue.
(iii) For each nonzero µ ∈ σ(T ), there is a smallest integer α, called ascent, with the property that of ker(µI − T )α = ker(µI − T )α+1 . Then
dim ker(µI −T )α is called the algebraic multiplicity of µ and dim ker(µI −
T ) is called the geometric multiplicity.
(iv) µ ∈ σ(T ) if and only if µ ∈ σ(T ∗ ). The ascent, algebraic multiplicity,
and geometric multiplicity of µ ∈ σ(T )\{0} and µ are equal, respectively.
The vectors in ker(µI − T ) are the eigenvectors associated with µ and those
in ker(µI − T )α are called generalized eigenvectors. Both ker(µI − T )α and
ker(µI − T ) are invariant under T . Note that the ascent of µI − T is one and
the two multiplicities are equal if T is self-adjoint; in this case the eigenvalues
are real (see Theorem A.76). Denoting by g the geometric multiplicity of µ,
it can be shown that α + g − 1 ≤ m ≤ αg.
Corollary A.75. Assume that dimL = ∞ and let T ∈ L(L; L) be a compact
operator. Then
(i) 0 ∈ σ(T ).
Part XII. Appendices
793
(ii) σ(T )\{0} = ε(T )\{0}.
(iii) One of the following situations holds: (1) σ(T ) = {0}; (2) σ(T )\{0} is
finite; (3) σ(T )\{0} is a sequence that converges to 0.
A.5.2 Symmetric operators in Hilbert spaces
Assume that L is a Hilbert space and let T ∈ L(L; L). The operator T is said
to be symmetric if (T v, w)H = (v, T w)H for all v, w ∈ H; equivalently, identifying L and L′ , T is self-adjoint, i.e., T = T ∗ . The key result for symmetric
operators is the following (see, e.g., [321, p. 356]).
Theorem A.76 (Real spectrum, spectral radius). Let L be a Hilbert
space and let T ∈ L(L; L) be a symmetric operator. Then, σ(T ) ⊂ R and
{a, b} ⊂ σ(T ) ⊂ [a, b],
(A.39)
with a = inf v∈H,kvkH =1 (T v, v)H and b = supv∈H,kvkH =1 (T v, v)H . Moreover,
kT kL(L;L) = |σ(T )| = max(|a|, |b|).
As a consequence of Corollary A.67, we infer the following
Corollary A.77 (Characterization of σ(T )). Let L be a Hilbert space and
let T ∈ L(L; L) be a symmetric operator. Then λ ∈ σ(T ) if and only if there is
a sequence (vn )n∈N in L such that kvn kL = 1 for all n ∈ N and T vn −λvn → 0
as n → ∞.
We conclude this section by considering symmetric compact operators.
Proposition A.78 (Symmetric compact operator). Let L be a Hilbert
space and let T ∈ L(L; L) be a symmetric compact operator. Then L has a
Hilbertian basis composed of eigenvectors of T .