More detailed problems are suggested in a separate section called "Problems" followed by "Partial solutions to the problems." The problems usually require knowledge of material that comes from different chapters. Some exercises and problems explain results stated without details or without proof in the main body of the chapter.
The Hahn–Banach Theorems. Introduction to the Theory of Conjugate Convex Functions
The Analytic Form of the Hahn–Banach Theorem: Extension of Linear Functionals
It is easy to see that the definition of h is logical, namely dath ∈ P, and dath is an upper bound on Q. Notation. We denote by the ethical space of E, that is, the space of all continuous linear functionals on E; the (double) standard onE is defined by.
The Geometric Forms of the Hahn–Banach Theorem
However, the "shoulder" in (5) is reached if it is a reflexive Banach space (see Chapter 3); a profound result due to R. James asserts the opposite: if it is a Banach space such that for every ∈Ethe sup in (5) is reached, then it is reflexive; see, e.g., J.
Separation of Convex Sets
The Bidual E . Orthogonality Relations
There is acanonic injection J :E → Defined as follows: given x ∈E, mapf → f, x is a continuous linear functional enE; thus it is an element of E, which we denote by J x.4We have. It is clear that J is linear and that J is anisometry, i.e. J xE = xE; we actually have.
A Quick Introduction to the Theory of Conjugate Convex Functions
In the general case it can be shown that (N⊥)⊥ coincides with the closure eN in the weak topology σ (E, E) (see Chapter 3). The classical form of Young's inequality (see the proof of Theorem 4.6 in Chapter 4) asserts this.
Comments on Chapter 1
- Generalizations and variants of the Hahn–Banach theorems
- Applications of the Hahn–Banach theorems
- Convex functions
- Extensions of bounded linear operators
This shows that the use of Zorn's lemma (and the underlying axiom of choice) in the proof of Hahn–Banach can be delicate and destroy the linear character of the problem. Damlamian [1] (for a problem arising in plasma physics), and by G. c) The theory of monotone operators and non-linear semigroups; see H. d) Variational problems involving periodic solutions of Hamiltonian systems and non-linear vibrating strings; see the recent works of F. e) The theory of large deviations in probability; see, e.g. R. f) The theory of partial differential equations and complex analysis; see L.
Exercises for Chapter 1
There exists a constant M≥0 such that for each finite subset J ⊂I and for each choice of real numbers(βi)i∈J, we have. To this end, let E be a Banach space and let G⊂Ebe a closed subspace that admits no complement (see Remark 8 in Chapter 2).
The Uniform Boundedness Principle and the Closed Graph Theorem
- The Baire Category Theorem
- The Uniform Boundedness Principle
- The Open Mapping Theorem and the Closed Graph Theorem
- Complementary Subspaces. Right and Left Invertibility of Linear Operators
- Orthogonality Revisited
- An Introduction to Unbounded Linear Operators. Definition of the Adjoint
- A Characterization of Operators with Closed Range
Sometimes one expresses the conclusion in Corollary 2.4 by saying that "weakly bounded"⇐⇒"strongly bounded" (see Chapter 3). By Hahn–Banach (analytic form; see Theorem 1.1) there exists a linear map f: E→R that extends and such.
A Characterization of Surjective Operators
However, it is always true that (A)⊥ is the closure of (A) for the weak topologyσ (E, E) (see Problem 9). This is similar to the proof of Theorem 2.20 and we will leave it as an exercise.
Comments on Chapter 2
It is easy to see that there is a bounded operator and that A =A;A (or A) is injective, but A (or A) is not surjective; RA)(or R(A)) is dense and not closed.
Exercises for Chapter 2
The Coarsest Topology for Which a Collection of Maps Becomes Continuous
Note that if we endow X with the discrete topology (ie, every subset of X is open), then every map ϕi is continuous; of course, this topology is far from "cheapest"; in fact it is the most expensive. As we will see, there is always a (unique) "cheapest" topology on X for which every map is continuous. The result is a family that is stable under the bottom; but it is not stable under ∪arbitrary.
Remark 2. Open (resp. closed) set in the weak topologyσ (E, E)areawaysopen (resp. closed) in the strong topology. In any infinite-dimensional space, the weak topology is more strictly coarse than the strong topology; i.e. there exist open (resp. closed) sets in the strong topology that are not open (resp. closed) in the weak topology. Remark 3. In infinite-dimensional spaces, the weak topology is never measurable, i.e. there is no metric (and a fortiori no norm) onEthat induces onE the weak topologyσ (E, E); see Exercise 3.8.
Weak Topology, Convex Sets, and Linear Operators
But as we shall see later (Theorem 3.29), if Eisseparable one can define a norm onEt that induces bounded sets of the weak topologyσ (E, E). Such spaces are quite "rare" and somewhat "pathological". This curious fact does not contradict Remark 2, which claims that in infinite dimensional spaces the weak topology and the strong topology are always distinct: the weak topology is strictly coarser than the strong topology. Dax belongs to the weak closure of∪∞p=1{xp}, it belongs a fortiori to the weak closure ofC.
Note 7. The argument above shows more: that if a linear operatorT is continuous from Estrong to F weak, then T is continuous from Estrong to F strong. There are two types of closed convex sets in E:. a) the convex sets that are strongly closed (=closed in the topology σ (E, E) by Theorem 3.7). Remark 12. The compactness of BE is the most important property of the weak topology; see also note 8.
Reflexive Spaces
It follows that J (BE) is not dense inBE in the strong topology, unless J (BE)=BE, i.e., reflexive Eis. Since J(E) is a closed subspace of the strong topology, we conclude (from Proposition 3.20) that J(E) is reflexive. 3 It is clear that if EandFare Banach space, and this is a linear surjective isometry from EontoF, then is reflexive iffFis reflexive.
Separable Spaces
Let f ∈ E be a continuous linear functional that vanishes on L; by Corollary 1.8, we must prove that f =0. The proof of the converse is more delicate (find where the proof above breaks down); we refer to N. Thus, we can find a subsequence (xnk) that converges weakly σ (M, M), and therefore (xnk) also converges weakly σ (E, E) (as in the proof of Proposition 3.20).
Uniformly Convex Spaces
Example 2. As we will see in Chapters 4 and 5, Lp spaces are uniformly convex for 1< p <∞and Hilbert spaces are also uniformly convex. On the other hand, reflexivity is an atopological property: a reflexive space remains reflexive to an equivalent norm. Uniform convexity is often used as a tool to prove reflexivity; but it is not the ultimate instrument; there are some strange reflexive spaces that do not admit a uniformly convex equivalent norm.
Comments on Chapter 3
The above references also include much material related to the Eberlein-Smulian theorem (Theorem 3.19).ˇ 3.The theory of vector spaces in duality – which extends the duality E, E – was very popular in the late forties and early fifties, especially in connection with the theory of distributions. These topologies are of interest in spaces that are not Banach spaces, such as the spaces used in distribution theory. For example, how does one know if a Banach space admits a corresponding uniformly convex norm.
Exercises for Chapter 3
Some Results about Integration That Everyone Must Know
A basic example is the case where = RN,M consists of the Lebesgue measurable sets, and μ is the Lebesgue measure on RN.
Definition and Elementary Properties of L p Spaces
The mappingφ →u, which is a linear surjective isometry, allows us to identify the "abstract" space (L1) with L∞. In what follows, we will systematically make the identification. Comment 6. The space L1() is never reflective, except in the trivial case where it consists of a finite number of atoms—and then L1() is finite-dimensional. Assuming that1 is reflexive, there exists a subsequence(enk) and someks∈1such that enk xin the weak topologyσ (1, ∞), i.e.,.
Convolution and regularization
Support and folding. The notion of supporting a function standard: suppf is the complement of the largest open set, where fvanishes; in other words suppf is the closure of the set {x;f (x) =0}. There is a countable family(Til) of open sets in RN such that every open set on RN is the union of some On's. Cc() is the space of continuous functions on with compact support in, i.e. which vanishes outside a compact setK.
Mollifiers
Criterion for Strong Compactness in L p
We recall that the Ascoli–Arzelà theorem answers the same question inC(K), the space of continuous functions over a compact metric spaceKwith values inR. Remark 11. When trying to establish that a familyFinLp() has compact closure inLp(), with bounded, it is usually convenient to extend the functions to the whole RN, then apply Theorem 4.26 and the restrictions to consider. Remark12.Under the assumptions of Theorem 4.26, we cannot in general conclude that Fitself has compact closure inLp(RN) (construct an example, or see Exercise 4.33).
Comments on Chapter 4
- Egorov’s theorem
- Weakly compact sets in L 1
- Radon measures
- The Bochner integral of vector-valued functions
- Interpolation theory
- Young’s inequality
ThenF has compact closure in the weak topologyσ(L1, L∞) if and only ifFis equi-integrable, that is,. Its dual space, denoted by M(), is called the Radon gauge space. The weak onM() topology is sometimes called the "fuzzy" topology. Since M() is the dual space of the divisible space C(), it has some compactness properties in the weak topology.
Exercises for Chapter 4
Hint: Given an integer ≥ 1, prove with the help of question 2 that there exists m⊂, measurable, such that |m|< δ/2and there exists an integer N such that. 6 One can show that (a) follows from (b) and (c) if the gauge space is diffuse (ie has no atoms).
Hilbert Spaces
The Dual Space of a Hilbert Space
Assume that V ⊂H is a linear subspace compacted in H. Assume that V has its norm and that V is a Banach space z. It is easy to see that T has the following properties:. iii)R(T ) is dense and VifV is reflexive.1. Remark 4. It is easy to prove that Hilbert spaces are reflexive without invoking the theory of uniformly convex spaces.
The Theorems of Stampacchia and Lax–Milgram
It follows that in a Hilbert space every closed subspace has a complement (in the sense of section 2.4). Using the Riesz-Fréchet theorem, we can now represent the functional ϕ via the new scalar product, that is, there exists a unique element g∈H such that. In the language of variational calculus, (17) is said to be the Euler equation related to the minimization problem (18).
Hilbert Sums. Orthonormal Bases
A sequence(s)n≥1inH is said to be anorthonormal basis ofH (or a Hilbert basis4 or simply a basis when there is no confusion)5 if it satisfies the following properties:. ii) the linear space spanned by theen's is dense inH. Note 9. The series in general. ukin Theorem 5.9 and the series. u, ek)ek in Corollary 5.10 is not absolutely convergent, i.e. it can happen that∞. Note 10. Theorem 5.11 combined with Corollary 5.10 shows that all separable Hilbert spaces are isomorphic and isometric with the space2.
Comments on Chapter 5
- Characterization of Hilbert spaces
- Variational inequalities
- Nonlinear equations associated with monotone operators
- Schauder bases in Banach spaces
Remark 11. If H is an inseparable Hilbert space – a rather unusual situation – one can still prove (using Zorn's lemma) the existence of an uncountable orthonormal basis (ei)i∈I; see for example W. Finally, let us recall a result already mentioned (Note 2.8). d) Theorem 5.15 (Lindenstrauss–Tzafriri [1]). Assume that E is a Banach space such that every closed subspace has a complement. 8 Then E is Hilbertizable, which means that there is an equivalent Hilbert norm. Other classical bases of L2(0,1) or L2(R) are associated with the names Bessel, Legendre, Hermite, Laguerre, Chebyshev, Jacobi, etc.
Exercises for Chapter 5
Prove that for eachf ∈Hthe seriesun=PKnf (strongly) converges to a limit and identify the limit. Prove that for everyf ∈ H the seriesun =PKnf (strongly) converges to a limit and identify the limit.
Compact Operators. Spectral Decomposition of Self-Adjoint Compact Operators
The Riesz–Fredholm Theory
Another linear result that has a simple proof based on Schauder's fixed point theorem is the Krein–Rutman theorem (see Theorem 6.13 and Problem 41). Then there exists a sequence (En) of finite-dimensional subspaces E such that En−1 ⊂ En and En−1 = En. Thus un-vn remains bounded, and since T is a compact operator, we can extract a subsequence such that T (unk-vnk) converges to some limit.
The Spectrum of a Compact Operator
On the other hand, R(I−T)= N (I−T)⊥ has a finite codimension (see Section 2.4, Example 2) and thus has a complement (inE), denoted by F, with dimension. In general, this inclusion may be strict:2 there may be several such. such belongs to the spectrum, but is not an eigenvalue). If we work in vector spaces over C (see Section 11.4) the situation is entirely different; the study of eigenvalues and spectra is much more interesting in spaces above C.
Spectral Decomposition of Self-Adjoint Compact Operators
If fu∈Emandv∈Enwithm =n, then T u=λmu and T v=λnv, so that. ii) Let F be a vector space spanned by spaces (En)n≥0. Finally, we choose in each subspace (En)n≥0a a Hilbert base (the existence of such a base for E0 follows from Theorem 5.11; for other Ens, n≥1, this is obvious, since they are finite-dimensional). Recall that in fact in Hilbert space every compact operator - not necessarily self-adjoint - is the limit of a sequence of operators of finite rank (see Note 1).
Comments on Chapter 6
Multiplicity of eigenvalues
Spectral analysis
The Krein–Rutman theorem
Exercises for Chapter 6
In what follows we assume that E is a Hilbert space (identified with its dual space H) and that T=T.
The Hille–Yosida Theorem
Definition and Elementary Properties of Maximal Monotone OperatorsOperators
Conclusion (c) follows easily by induction: since I+Ais surjective, I+λAis surjective for everyλ >1/2, and therefore for everyλ >1/4, etc.
