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Spectral Analysis of Linear Operators
SMA 5878 Functional Analysis II
Alexandre Nolasco de Carvalho
Departamento de MatematicaInstituto de Ciencias Matematicas and de Computacao
Universidade de Sao Paulo
March 25, 2019
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsSymmetric and self-adjoint operators
Min-Max Characterisation of Eigenvalues
Symmetric and self-adjoint operators
Let H be a Hilbert space with inner product 〈·, ·〉H : H × H → K
and A : D(A) ⊂ H → H be a densely defined operator. Theadjoint operator A• of A is defined by
D(A•) = {u ∈ H : v 7→ 〈Av , u〉H : D(A) → K is bounded}
and if u ∈ D(A•), A•u is the only element of H such that
〈v ,A•u〉H = 〈Av , u〉H ,∀v ∈ D(A).
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsSymmetric and self-adjoint operators
Min-Max Characterisation of Eigenvalues
RemarkIf H is a Hilbert space over C, E : H → H∗ defined byEu(v) = 〈v , u〉H , is a conjugated isometry between H and H∗.The identification between H and H∗ consists in identifying u withEu. If A∗ : D(A∗) ⊂ X ∗ → X ∗ is the dual of A, thenA• = E−1 ◦ A∗ ◦ E.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsSymmetric and self-adjoint operators
Min-Max Characterisation of Eigenvalues
RemarkAlso note that, even though E and E−1 be linear conjugatedoperators, E−1 ◦ A∗ ◦ E is a linear operator by double conjugation.We cal both A• and A∗ ajoints A and we denote both by A∗ but itis important to note that, if A = αB then A• = αB• while thatA∗ = αB∗. From this, (λI−A)•= λI−A• while (λI−A)∗=λI ∗−A∗.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsSymmetric and self-adjoint operators
Min-Max Characterisation of Eigenvalues
If no confusion may arise we will use the notation A∗ to denote thedual and the adjoint operators, indistinctively. In that case we mayalso refer to both as the adjoint operator.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsSymmetric and self-adjoint operators
Min-Max Characterisation of Eigenvalues
DefinitionLet H be a Hilbert space over K with inner product 〈·, ·〉. We saythat an operator A : D(A) ⊂ H → H is symmetric (also calledHermitian when K = C) if D(A) = H and A ⊂ A•; that is,〈Ax , y〉 = 〈x ,Ay〉 for all x , y ∈ D(A). We say that A is self-adjointif A = A•.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsSymmetric and self-adjoint operators
Min-Max Characterisation of Eigenvalues
ExerciseLet H is a Hilbert space. If A : D(A) ⊂ H → H is a denselydefined operator, then A• : D(A•) ⊂ H → H is closed. Besidesthat, if A is closed, then A• is densely defined.
ExerciseLet H be a Hilbert space over K. Show that, ifA : D(A) ⊂ H → H is symmetric and λ ∈ K is an eigenvalue of A,then λ ∈ R. Besides that,
inf‖x‖H=1
〈Ax , x〉 ≤ λ ≤ sup‖x‖H=1
〈Ax , x〉.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsSymmetric and self-adjoint operators
Min-Max Characterisation of Eigenvalues
ExerciseLet H = C
n with the usual inner product. If A = (ai , j)ni , j=1 is a
matrix with complex coefficients that represents a linear operatorA ∈ L(H), find A• and A∗.
ExerciseLet H be a Hilbert space over K with inner product 〈·, ·〉 andA : D(A) ⊂ H → H is a densely defined operator. Show thatG (A•) = {(−Ax , x) : x ∈ D(A)}⊥ (here M⊥ represents theortogonal M).
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsSymmetric and self-adjoint operators
Min-Max Characterisation of Eigenvalues
Proposition
Let H be a Hilbert space over K with inner product 〈·, ·〉. IfA : D(A) ⊂ H → H is a self-adjoint operator, which is injectiveand with dense image, then A−1 is self-ajoint.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsSymmetric and self-adjoint operators
Min-Max Characterisation of Eigenvalues
Proof: Since A is self-adjoint, it is easy to see that
{(x ,−Ax) : x ∈ D(A)}⊥ = {(Ax , x) : x ∈ D(A)} = G (A−1).
Since A is injective and has dense image, it follows from theprevious exercise,
G ((A−1)•) = {(−A−1x , x) : x ∈ R(A)}⊥ = G (A−1).
Hence A−1 = (A−1)•.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsSymmetric and self-adjoint operators
Min-Max Characterisation of Eigenvalues
TheoremLet H be a Hilbert space over K with inner product 〈·, ·〉. IfA : D(A) ⊂ H → H is a symmetric and surjective linear operator,then A is self-adjoint.
Proof: First we show that A and A∗ are both injective. Ifx ∈ D(A) and Ax = 0, we have that 〈Ax , y〉 = 〈x ,Ay〉 for ally ∈ D(A). Consequently, from the fact that R(A) = X , we havethat x = 0. To see that A∗ is injective we proceed the same way.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsSymmetric and self-adjoint operators
Min-Max Characterisation of Eigenvalues
Now we show that A is closed. In fact, ifD(A∗) ⊃ D(A) ∋ xn → x ∈ X and Axn = A∗xn → y , thenx ∈ D(A∗) and A∗x = y . Since A is surjective, there existsw ∈ D(A) such that Aw = A∗w = A∗x and from the injectivity ofA∗ we have that w = x . With this x ∈ D(A) and Ax = y , showingthat A is closed.
It follows from the Closed Graph Theorem that a A has inverseA−1 ∈ L(X ). Clearly A−1 is self-adjoint and, from a previousresult, it follows that A is self-adjoint.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsSymmetric and self-adjoint operators
Min-Max Characterisation of Eigenvalues
Our next theorem and the previous theorem are the main tools toshow that a linear operator is symmetric.
Theorem (Friedrichs)
Let X be a Hilbert space over K and A : D(A) ⊂ X → X asymmetric operator for which there exists a α ∈ R such that
〈Ax , x〉 ≤ α‖x‖2 or 〈Ax , x〉 ≥ α‖x‖2 (1)
for all x ∈ D(A). Then A admits a self-adjoint extension thatpreserves the bound (1).
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsSymmetric and self-adjoint operators
Min-Max Characterisation of Eigenvalues
Proof: We prove only the case 〈Ax , x〉 ≥ α‖x‖2 for all x ∈ D(A)and for some α ∈ R. The remaining case can be deduced from thisconsidering the operator −A.
Also, we only consider the case α = 1 for the general case can bededuced from this considering the operator A+ (1− α)I .
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsSymmetric and self-adjoint operators
Min-Max Characterisation of Eigenvalues
In D(A) consider the inner productD(A)× D(A) ∋ (x , y) 7→ 〈Ax , y〉 ∈ K. Clearly the norm
D(A) ∋ x 7→ ‖x‖ 12= 〈Ax , x〉
12 ∈ R
+ from this inner product
satisfies ‖x‖ 12≥ ‖x‖. Denote by X
12 the completion of D(A)
relatively to the norm ‖ · ‖ 12.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsSymmetric and self-adjoint operators
Min-Max Characterisation of Eigenvalues
Let us show that X12 , as a set, is in a bijective correspondance with
a subset of the completion of D(A) relatively to the norm ‖ · ‖. Itis clear that all sequences {xn} in D(A) that are Cauchy relativelyto the norm ‖ · ‖ 1
2is also Cauchy relatively to the norm ‖ · ‖.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsSymmetric and self-adjoint operators
Min-Max Characterisation of Eigenvalues
To conclude the injectivity we show, by reduction to absurd, that if{xn} is a Cauchy sequence relatively to the norm ‖ · ‖ 1
2and for
which limn→∞ ‖xn‖ 12= a > 0, we cannot have that
limn→∞ ‖xn‖ = 0.
If the thesis is false, we have that
2Re〈Axn, xm〉 = 〈Axn, xn〉+ 〈Axm, xm〉 − 〈A(xn − xm), (xn − xm)〉m,n→∞−→ 2a2
which is an absurd since 〈Axn, xm〉m→∞−→ 0.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsSymmetric and self-adjoint operators
Min-Max Characterisation of Eigenvalues
Since X is complete, X12 can be identified with a subset of X .
Let D = D(A∗) ∩ X12 . Since D(A) ⊂ D(A∗), we must have that
D(A) ⊂ D ⊂ D(A∗). We define A taking the restriction of A∗ to Dand showing that A is self-adjoint.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsSymmetric and self-adjoint operators
Min-Max Characterisation of Eigenvalues
Let us first show that A is symmetric. If x , y ∈ D there aresequences {xn} and {yn} in D(A) such that ‖xn − x‖ 1
2
n→∞−→ 0
‖yn − y‖ 12
n→∞−→ 0.
It follows that
limm→∞
limn→∞
〈Axn, ym〉 = limn→∞
limm→∞
〈Axn, ym〉 = 〈x , y〉 12
exists and coincides with
limn→∞
limm→∞
〈Axn, ym〉 = limn→∞
〈Axn, y〉 = limn→∞
〈xn, Ay〉 = 〈x , Ay〉 and with
limm→∞
limn→∞
〈Axn, ym〉 = limm→∞
〈x ,Aym〉 = limm→∞
〈Ax , ym〉 = 〈Ax , y〉.
Hence A is symmetric.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsSymmetric and self-adjoint operators
Min-Max Characterisation of Eigenvalues
To conclude the proof it is enough to show that A is surjective andthis is done in the following way.
Let y ∈ X and consider the linear functional f : D(A) → K givenby f (x) = 〈x , y〉.
Then f is a bounded linear functional with respect to the norm‖ · ‖ 1
2and can be extended to a bounded linear functional defined
in X12 . From Riesz Representation Theorem, there exists y ′ ∈ X
12
such that
f (x) = 〈x , y〉 = 〈x , y ′〉 12= 〈Ax , y ′〉, ∀x ∈ D(A).
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsSymmetric and self-adjoint operators
Min-Max Characterisation of Eigenvalues
Hence y ′ ∈ D(A∗) ∩ X12 and A∗y ′ = Ay ′ = y showing that A is
surjective. From here we also conclude that (−∞, 1) ⊂ ρ(A).
Since A is symmetric and surjective, it follows that A is self-adjointand is an extension of A. It is easy to see that 〈Ax , x〉 ≥ ‖x‖2 forall x ∈ D(A). This concludes the proof.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsSymmetric and self-adjoint operators
Min-Max Characterisation of Eigenvalues
Example
Let X = L2(0, π) and D(A0) = C 20 (0, π) the set of functions which
are twice continuously differentiable functions and have compactsupport in (0, π). Define A0 : D(A0) ⊂ X → X by
(A0φ)(x) = −φ′′(x), x ∈ (0, π).
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsSymmetric and self-adjoint operators
Min-Max Characterisation of Eigenvalues
It is easy to see that A0 is symmetric and that 〈A0φ, φ〉 ≥2π2 ‖φ‖
2X
for all φ ∈ D(A0).
From Theorem 2, A0 has a self-adjoint extension A that satisfies〈Aφ, φ〉 ≥ 2
π2 ‖φ‖2X for all φ ∈ D(A).
Note that, the space X12 from Friedrichs theorem is, in this
example the closure of D(A) in the norm H1(0, π) and therefore
X12 = H1
0 (0, π).
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsSymmetric and self-adjoint operators
Min-Max Characterisation of Eigenvalues
On the other hand D(A∗) is characterised by
D(A∗0) = {φ ∈ X : ∃φ∗ ∈ X such that 〈−u′′, φ〉 = 〈u, φ∗〉, ∀u ∈ D(A0)}
and A∗0φ=−φ′′ for all φ∈D(A∗
0).
Hence, D(A)=H2(0, π)∩H10 (0, π) and Au=−u′′ for all u∈D(A).
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsSymmetric and self-adjoint operators
Min-Max Characterisation of Eigenvalues
Also from Theorem 2 we know that (−∞, 2π2 ) ⊂ ρ(A). In
particular 0 ∈ ρ(A) and if φ ∈ D(A), we have that
|φ(x)− φ(y)| ≤ |x − y |12‖φ′‖L2(0,π) = |x − y |
12 〈Aφ, φ〉
12 .
Hence, if B is a bounded subset of D(A) with the graph norm,then supφ∈B ‖φ′‖L2(0,π) < ∞ and the family B of functions isequicontinuous and bounded in C ([0, π],R) with the uniformconvergence topology.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsSymmetric and self-adjoint operators
Min-Max Characterisation of Eigenvalues
It follows from the Arzela-Ascoli Theorem that B is relativelycompact in C ([0, π],R) and consequently B is relatively compactin L2(0, π).
From a previous exercise it follows that A−1 is a compact operator.
It follows that σ(A) = {λ1, λ2, λ3, · · · } where λn = n2 ∈ σp(A)
with eigenfunctions φn(x) =(
2π
)12 sen(nx), n ∈ N.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsSymmetric and self-adjoint operators
Min-Max Characterisation of Eigenvalues
Min-Max Characterisation of Eigenvalues
In this section we introduce min-max characterisations ofeigenvalues of compact and self-adjoint operators. To presentthese characterisations we will employ the following result
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsSymmetric and self-adjoint operators
Min-Max Characterisation of Eigenvalues
LemmaLet H be a Hilbert space over K and A ∈ L(H) be a self-adjointoperator, then
‖A‖L(H) = sup‖u‖=1‖v‖=1
|〈Au, v〉| = sup‖u‖=1
|〈Au, u〉|.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsSymmetric and self-adjoint operators
Min-Max Characterisation of Eigenvalues
Proof: It is enough to prove that
‖A‖L(H) = sup‖u‖=1‖v‖=1
|〈Au, v〉| ≤ sup‖u‖=1
|〈Au, u〉| := a.
If u, v ′ ∈ H, ‖u‖ = ‖v ′‖ = 1, |〈Au, v ′〉| e iα = 〈Au, v ′〉 andv = e−iαv ′, we have that
|〈Au, v ′〉| = 〈Au, v〉 =1
4[〈A(u + v), u + v〉 − 〈A(u − v), u − v〉]
≤a
4[‖u + v‖2 + ‖u − v‖2] ≤ a.
This completes the proof.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsSymmetric and self-adjoint operators
Min-Max Characterisation of Eigenvalues
ExerciseShow that, if 0 6= A ∈ L(H) is self-adjoint, then A is notquasinilpotent.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsSymmetric and self-adjoint operators
Min-Max Characterisation of Eigenvalues
TheoremLet H be a Banach space over K and A ∈ K(H) be a self-adjointoperator such that 〈Au, u〉 ≥ 0 for all u ∈ H. Then,
1. λ1 :=sup{〈Au, u〉 :‖u‖=1} is an eigenvalue and exists v1∈H,‖v1‖=1 such that λ1=〈Av1, v1〉. Besides that Av1=λ1v1.
2. Inductively,
λn :=sup{〈Au, u〉 :‖u‖=1 and u⊥vj , 1≤ j≤n−1} ∈σp(A) (1)
and exists vn ∈ H, ‖vn‖ = 1, vn ⊥ vj , 1 ≤ j ≤ n − 1, suchthat λn = 〈Avn, vn〉. Besides that Avn = λnvn.
3. If Vn = {F ⊂ H : F is a vec. subesp. of dimension n de H},
λn = infF∈Vn−1
sup{〈Au, u〉 : ‖u‖ = 1, u ⊥ F}, n ≥ 1 and (2)
λn = supF∈Vn
inf{〈Au, u〉 : ‖u‖ = 1, u ∈ F}, n ≥ 1. (3)
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsSymmetric and self-adjoint operators
Min-Max Characterisation of Eigenvalues
Proof: We consider only the case K = C and λ1 > 0 leaving theremaining cases as exercises for the reader.
1.Let {un} be a sequence in H with ‖un‖=1 and 〈Aun, un〉n→∞−→ λ1.
Taking subsequences if necessary, {un} converges weakly to v1 ∈ Hand {Aun} converges strongly to Av1.
Hence 〈Av1, v1〉 = λ1.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsSymmetric and self-adjoint operators
Min-Max Characterisation of Eigenvalues
Let us show that the sequence {un} converges strongly.
From the previous lemma we know that 0 < λ1 = ‖A‖L(H) andfrom the fact that {un} converges weakly to v1 we have that0 < ‖v1‖ ≤ 1. Hence,
limn→∞
‖Aun − λ1un‖2 = lim
n→∞‖Aun‖
2 − 2λ1 limn→∞
〈Aun, un〉+ λ21
= ‖Av1‖2 − λ2
1 ≤ 0.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsSymmetric and self-adjoint operators
Min-Max Characterisation of Eigenvalues
Since {Aun} converges strongly to Av1, {Aun − λ1un} convergesstrongly to zero and λ1 > 0, it follows that {un} converges stronglyto v1, ‖v1‖ = 1 and Av1 = λ1v1. This concludes the proof of 1.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsSymmetric and self-adjoint operators
Min-Max Characterisation of Eigenvalues
2. The proof of this item follows from 1. simply noting that theorthogonal of Hn−1 = span{v1, · · · , vn−1} is invariant by A andrepeating the procedure for the restriction of A to H⊥
n−1, n ≥ 2.This concludes the proof of 2.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsSymmetric and self-adjoint operators
Min-Max Characterisation of Eigenvalues
3. We first prove expression (2). If G = span{v1, · · · , vn−1} wehave, from (1), that
λn = sup{〈Au, u〉 : ‖u‖ = 1, u ⊥ G}
≥ infF∈Vn−1
sup{〈Au, u〉 : ‖u‖ = 1, u ⊥ F}.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsSymmetric and self-adjoint operators
Min-Max Characterisation of Eigenvalues
On the other hand, let F ∈ Vn−1 and w1, · · · ,wn−1 anorthornormal set of F . Choose u =
∑ni=1 αivi such that ‖u‖ = 1
and u ⊥ wj , 1 ≤ j ≤ n − 1. Hence∑n
i=1 |αi |2 = 1 and
〈Au, u〉 =
n∑
i=1
|αi |2λi ≥ λn.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsSymmetric and self-adjoint operators
Min-Max Characterisation of Eigenvalues
This implies
sup{〈Au, u〉 : ‖u‖ = 1, u ⊥ F} ≥ λn, for all F ∈ Vn−1
and completes the proof of (2).
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsSymmetric and self-adjoint operators
Min-Max Characterisation of Eigenvalues
We now prove (3). If G = span{v1, · · · , vn} and u ∈ G , ‖u‖ = 1,we have that u =
∑ni=1 αivi with
∑ni=1 |αi |
2 = 1 e
〈Au, u〉 =
n∑
i=1
|αi |2λi ≥ λn.
This implies that
supF∈Vn
inf{〈Au, u〉 : ‖u‖ = 1, u ∈ F} ≥ λn.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsSymmetric and self-adjoint operators
Min-Max Characterisation of Eigenvalues
Conversely, given F ∈ Vn choose u ∈ F , ‖u‖ = 1, such thatu ⊥ vj , 1 ≤ j ≤ n − 1. It follows, from 2., that 〈Au, u〉 ≤ λn andconsequently
inf{〈Au, u〉 : ‖u‖ = 1, u ∈ F} ≤ λn, for all F ∈ Vn.
This completes the proof of (3).
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Spectral Analysis of Linear OperatorsSymmetric and self-adjoint operators
Min-Max Characterisation of Eigenvalues
ExerciseIf A : D(A) ⊂ H → H is positive, self-adjoint and (〈Au, u〉 > 0 forall u ∈ D(A)) and has compact resolvent, find the min-maxcharacterisation for the eigenvalues of A.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II