Download Mathematical Functional Analysis Class notes and more Lecture notes Mathematics in PDF only on Docsity!
GIM/ DWD 2011 1
MTH 3101: FUNCTIONAL ANALYSIS^1
G.I. Mirumbe & D. W. Ddumba 2
References: Peter D. Lax, Functional Analysis, Courant institute, New York University, 2002. Erwin Kryszig, Introductory functional analysis, 1978. John B. Conway, A Course in Functional Analysis, 2nd Edition, Springer-Verlag, 1990. Gert K. Pedersen, Analysis Now, Springer-Verlag, 1989. Walter Rudin, Functional Analysis, 2nd Edition, McGraw Hill, 1991. Robert J. Zimmer, Essential Results of Functional Analysis, University of Chicago Press, 1990. (^1) These lecture notes were prepared for the instructor’s personal use in teaching a semester course on functional analysis at the undergraduate level in August 2011. They are certainly not meant to replace a good text on the subject such as those listed on this page and also should not be used as a replacement for the lectures. (^2) Department of Mathematics, (Mak).
Web: http://www.math.mak.ac.ug
Contents
Chapter 1
Introduction
Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of vector spaces and functions/ maps acting on them. It has its historical roots in the study of functional spaces, in particular transformations of functions, such as the Fourier transform, as well as in the study of differential and integral equations. This usage of the word functional goes back to the calculus of variations, implying a function whose argument is a function.
In the modern view, functional analysis is seen as the study of complete normed vector spaces over the space of real or complex numbers. Such spaces are called Banach spaces. An important example is a Hilbert space, where the norm arises from an inner product. These spaces are of fundamental importance in many areas, including the mathematical formulation of quantum mechanics. More generally, functional analysis includes the study of Fr´echet spaces and other topological vector spaces not endowed with a norm but in- stead endowed with a sequence of semi-Norms.
An important object of study in functional analysis are the continuous linear operators defined on Banach and Hilbert spaces which lead naturally to the definition of C∗-algebras and other operator algebras.
For this course we want to catagorise functions, whether some functions are Hilbert or Banach spaces or neither. To be able to determine, we do check whether they have the inner-products, and to determine whether some functions have inner products is to determine whether they have semi inner-products. That is
Semi-Inner Product ⇓ Inner Product ⇒ Norm ↙ ⇓ Hilbert Spaces ⇒ Banach Spaces
Chapter 2
Metric spaces
2.1 Introduction
If x, y ∈ X, then the distance between these points is denoted by d(x; y). The function d(−; −) : X × X −→ R has to satisfy several conditions before the function d(x, y) is called a distance function on X or a metric on X.
Definition 2.1.1. Metric: Suppose that X is a non-empty set and d : X × X :→ R is said to be a metric if it satisfies the following conditions,
(i) d(x, y) ≥ 0 for all x, y ∈ X.
(ii) d(x, y) = 0 if and only if x = y.
(iii) d(x, y) = d(y, x) for all x, y ∈ X. (Symmetry).
(iv) d(x, y) + d(y, z) ≥ d(x, z) for all x, y, z ∈ X.
[The forth condition is called the ‘triangle inequality’ after the result in Euclidean geom- etry that the sum of the lengths of two sides of a triangle is at least as great as the length of the third side].
Definition 2.1.2. The pair (X, d) is called the metric space.
You should imagine the lecturer muttering under his breath
(i) Distances are always positive.
(ii) Two points are zero distance apart if and only if they are the same point.
(iii) The distance from A to B is the same as the distance from B to A.
(iv) The distance from A to B via C is at least as great as the distance from A to B directly.
For most of the course we shall be concerned with metrics which you already know well.
GIM/ DWD 2011 7
2.3 Some properties of the Metric spaces
The next definitions work in any metric space but, if you have not met them before, you can concentrate on what they mean for ordinary Euclidean space.
Definition 2.3.1. If (X, d) is a metric space xn ∈ X, x ∈ X and d(xn, x) → 0, then we say that xn → x as n → ∞.
Definition 2.3.2. If (X, d) is a metric space x ∈ X and r > 0, then we write
B(x 0 , r) = {x ∈ X : d(x, x 0 ) < r}
and call B(x 0 , r) the open ball of radius r with center x 0 and if
B(x 0 , r) = {x ∈ X : d(x, x 0 ) ≤ r}
then it is a closed ball of radius r.
Definition 2.3.3. Let Y and X be sets and Y ⊆ X. Y is a dense subset of X, if for every x ∈ X, there exists a sequence of elements {yn} n ∈ N in Y , such that lim n→∞
yn = x.
It can also be said that Y is a dense subset of X if Y = X. The rational numbers Q is a dense subset of real numbers R, Q lies dense in R.
Definition 2.3.4. Let (X, d) be a metric space.
(i) We say that a subset E of X is closed if, whenever xn ∈ E and xn → x, it follows that x ∈ E.
(ii) We say that a subset U of X is open if, whenever u ∈ U, we can find a δ > 0 such that B(u, δ) ⊆ U.
(iii) (X, d) is a compact Metric Space if every sequence in X has a subsequence that converges to a point in X. An example of a compact metric space is a bounded and closed interval [a, b], a, b ∈ R with the metric d(x; y) = |x − y|.
We recall (without proof) the following important results from Real analysis I,
Theorem 2.3.1. [Cauchy criterion]: A sequence an ∈ R converges if and only if, given > 0, we can find an N() such that |an − am| < for all n, m ≥ N()
Definition 2.3.5. If (X, d) is a metric space, then a sequence (an) with an ∈ X is called a Cauchy sequence if, given > 0, we can find an N() such that d(an, am) < for all n, m ≥ N().
Definition 2.3.6. We say that a metric space (X, d) is complete if every Cauchy sequence in X converges to an element x ∈ X. We note the following useful remarks.
8 Functional analysis lecture Notes
Lemma 2.3.1. Let (X, d) be a metric space.
(i) If a Cauchy sequence xn in (X, d) has a convergent subsequence with limit x then xn → x.
(ii) Let n > 0 and n → 0 as n → ∞. If (X, d) has the property that, whenever d(xn, xn+1) < n for n ≥ 1, it follows that the sequence xn converges, then (X, d) is complete.
Lemma 2.3.1 (ii) is most useful when we have
n=1 n^ convergent, for example if^ n^ = 2
−n (^).
Definition 2.3.7. Let G be some subset of X. x ∈ G is called an interior point of G, if there exists some r > 0, such that B(x 0 , r) is fully contained in G. The set of all interior points of G is called the interior of G and is denoted by G◦. We now come to our first real theorems.
Theorem 2.3.2. The distance function d(·, ·) : X × X −→ R is continuous.
Theorem 2.3.3. The Euclidean space Rm^ with the usual metric is complete.
Theorem 2.3.4. For 1 ≤ p < ∞, the metric space lp^ is complete.
Note 2.3.1. The distance function d could as well be used to define the distance between a point and a set, the distance between two sets and the diameter of a set.
Definition 2.3.8. Let (X; d) be a metric space and the sets A and B are non-empty subsets of X.
(a) The distance between a point x ∈ X and a set A ⊂ X is denoted and defined by
dist(x; A) = inf {d(x; y)|y ∈ A}
(b) The distance between the sets A ⊂ X and B ⊂ X is denoted and defined by
dist(A; B) = inf {d(x; y)|x ∈ A; y ∈ B}
(c) The diameter of A ⊂ X is denoted and defined by
diam(A) = sup {d(x; y)|x ∈ A; y ∈ A}
Exercise 2.1.
- Show that if (X, d) is a metric space (complete or not), then every convergent sequence is Cauchy.
- Proof lemma 2.2.1.
Chapter 3
Inner product spaces
3.1 Introduction
The basic algebraic concepts in the theory of Hilbert spaces are those of a vector space and an inner product. The inner product induces a norm and thus every Hilbert space is a normed space. Since the norm plays a very important role in the theory, it is not possible to study Hilbert spaces without familiarity with basic concepts and properties of normed spaces. We will consider both real vector spaces and complex vector spaces. The field of real numbers will be denoted by R and the field of complex numbers by C. Elements of R or C are called scalars. Sometimes it is convenient to give a definition or state a theorem without specifying the field of scalars. In such a case we will use F to denote either R or C.
3.2 Some definitions
Definition 3.2.1. Semi-inner product: Let X be a vector space over a field F. A semi-inner product U, is a function
U : X × X → F (R or C)
such that ∀ α, β ∈ F with x, y, z ∈ X the following properties are satisfied.
(a) U (αx + βy, z) = αU(x, z) + βU(y, z)
(b) U (x, αy + βz) = αU(x, y) + βU(x, z), where α and β are the usual complex conju- gates.
(c) U (x, x) ≥ 0
(d) U (x, y) = U(y, x)
(e) U (x, 0) = U(0, y) = 0 ∀ x, y ∈ X. In particular U(0, 0) = 0
GIM/ DWD 2011 11
Definition 3.2.2. An inner product on X is a semi-inner product that on top of satisfying a to e also satisfies
(f) If U (x, x) = 0 then x = 0.
Notation 3.2.1. : we write 〈x, y〉 = U(x, y) if U is an inner product.
Note 3.2.1. U(x, y) is a constant (scalar) which could be a complex number or a real number.
Definition 3.2.3. The pair (X, < .,. >) is the inner product space or some times called the Unitary space or a pre-Hilbert space.
3.3 Examples of Inner products and semi inner prod-
ucts
- If z 1 and z 2 denotes complex numbers, then 〈z 1 , z 2 〉 = z 1 z 2 , defines an inner product on the complex plane Cn.
- Let X be a collection of all sequences {αn : n ≥ 1 } of scalars αn ∈ F such that αn = 0 for all but a finite number of values of n.
Define addition and scalar multiplication by
{αn} + {βn} = {αn + βn} c{αn} = {cαn}
Then X is a vector space (it satisfies all properties of vector spaces). But for an inner product, if
U ({αn}, {βn}) =
∑^ ∞
n=
α 2 nβ 2 n
Then U is a semi-inner product that is not an inner product.
For example let x ∈ X be defined as
x = (1, 0 , 1 , 0 , 1 , 0 , · · · )
Then
αn = (1, 1 , 1 , 1 , · · · ) βn = (1, 1 , 1 , 1 , · · · ) α 2 n = (0, 0 , 0 , 0 , · · · ) β 2 n = (0, 0 , 0 , 0 , · · · )
GIM/ DWD 2011 13
Proof. If α ∈ F , x, y ∈ X then
0 ≤ 〈x − αy, x − αy〉 since 〈x, x〉 ≥ 0 0 ≤ 〈x, x〉 + 〈x, −αy〉 + 〈−αy, x〉 + 〈−αy, −αy〉 0 ≤ 〈x, x〉 − α 〈x, y〉 − α 〈y, x〉 + |α|^2 〈y, y〉 0 ≤ 〈x, x〉 − α 〈y, x〉 − α 〈y, x〉 + |α|^2 〈y, y〉 0 ≤ 〈x, x〉 − 2 Re (α 〈y, x〉) + |α|^2 〈y, y〉 since ¯z + z = 2Re(z)
Since 〈·, ·〉 is a scalar, let it be a complex number z where |z| = b and arg z = θ. Define
〈y, x〉 = beiθ^ , b = | 〈y, x〉 | ≥ 0 α = te−iθ^ , t ∈ R
0 ≤ 〈x, x〉 − 2 Re
te−iθ^ · beiθ
- t^2 〈y, y〉 0 ≤ 〈x, x〉 − 2 Re (bt) + t^2 〈y, y〉 0 ≤ 〈x, x〉 − 2 bt + t^2 〈y, y〉
A quadratic polynomial in t Thus the equation
0 = 〈x, x〉 − 2 bt + t^2 〈y, y〉
has at most one solution of t, thus b^2 − 4 ac ≤ 0. Since a = 〈y, y〉 , b = − 2 b, c = 〈x, x〉
4 b^2 − 4 〈x, x〉 〈y, y〉 ≤ 0 b^2 − 〈x, x〉 〈y, y〉 ≤ 0 | 〈y, x〉 |^2 − 〈x, x〉 〈y, y〉 ≤ 0 | 〈x, y〉 |^2 − 〈x, x〉 〈y, y〉 ≤ 0 since | 〈y, x〉 | = | 〈x, y〉 | | 〈x, y〉 |^2 ≤ 〈x, x〉 〈y, y〉
Note 3.4.1. This theorem can also be proved as follows
If α ∈ F , x, y ∈ X then
0 ≤ 〈x − αy, x − αy〉 , since 〈x, x〉 ≥ 0 0 ≤ 〈x, x〉 + 〈x, −αy〉 + 〈−αy, x〉 + 〈−αy, −αy〉
0 ≤ 〈x, x〉 − α 〈x, y〉 − α 〈y, x〉 + αα 〈y, y〉 (3.2)
If y = 0 then the expression leads to the property 0 ≤ 〈x, x〉. Suppose instead that y 6 = 0 and subsequently < y, y > 6 = 0, Now letting
α =
< x, y > < y, y >
then substituting this into 3.2 yields the requested inequality.
14 Functional analysis lecture Notes
3.4.1 Corollary
If 〈·, ·〉 is an inner-product on X and
‖x‖ = 〈x, x〉
(^12)
for all x in X, then
(a) ‖x + y‖ ≤ ‖x‖ + ‖y‖, ∀ x, y ∈ X
Proof.
‖x + y‖^2 = 〈x + y, x + y〉 = 〈x, x〉 + 〈x, y〉 + 〈y, x〉 + 〈y, y〉 = 〈x, x〉 + 〈x, y〉 + 〈x, y〉 + 〈y, y〉 = 〈x, x〉 + 2Re (〈x, y〉) + 〈y, y〉 since z + ¯z = 2Re(z) = ‖x‖^2 + 2Re (〈x, y〉) + ‖y‖^2 ≤ ‖x‖^2 + 2| 〈x, y〉 | + ‖y‖^2 since Re(z) ≤ |Re(z)| ≤ |z|
From Cauchy Schwarz inequality
‖x + y‖^2 ≤ ‖x‖^2 + 2 〈x, x〉
(^12) 〈y, y〉
(^12)
- ‖y‖^2 ‖x + y‖^2 ≤ ‖x‖^2 + 2‖x‖‖y‖ + ‖y‖^2 ‖x + y‖^2 ≤ (‖x‖ + ‖y‖)^2
Taking the square roots on both sides, we get the result.
(b) ‖αx‖ = |α|‖x‖ for α ∈ F , x ∈ X
Proof.
‖αx‖^2 = 〈αx, αx〉 ‖αx‖^2 = |α|^2 〈x, x〉 ‖αx‖^2 = |α|^2 ‖x‖^2
Taking square roots on both sides
‖αx‖ = |α|‖x‖
(c) ‖x‖ = 0 ⇒ x = 0
16 Functional analysis lecture Notes
Theorem 3.6.1. Given x ⊥ y then ‖x|^2 + ‖y‖^2 = ‖x + y‖^2
Proof.
‖x + y‖^2 = 〈x + y, x + y〉 = 〈x, x〉 + 〈x, y〉 + 〈y, x〉 + 〈y, y〉 = 〈x, x〉 + 0 + 0 + 〈y, y〉 since orthogonal 〈x, y〉 = 〈y, x〉 = 0 = 〈x, x〉 + 〈y, y〉 = ‖x‖^2 + ‖y‖^2
Theorem 3.6.2. An inner product < .,. > can be recovered by the norm ‖.‖ on a Vector Space X if and only if the norm ‖.‖ satisfies the parallelogram identity
2 ‖x‖^2 + 2‖y‖^2 = ‖x + y‖^2 + ‖x − y‖^2
The inner product is given by the polarization identity
< x, y >=
‖x + y‖^2 − ‖x − y‖^2 ) + i(‖x + iy‖^2 − ‖x − iy‖^2 )
3.7 Continuity of an Inner-Product
If in an inner product space xn → x and yn → y then
〈xn, yn〉 → 〈x, y〉
Recall 3.1. In calculus if xn → x then f(xn) → f(x).
Proof. Note that if xn → x iff ‖xn − x‖ → 0 and yn → y iff ‖yn − y‖ → 0, thus we are required to prove that ‖〈xn, yn〉 − 〈x, y〉‖ → 0
‖〈xn, yn〉 − 〈x, y〉‖ = ‖〈xn, yn〉 − 〈xn, y〉 + 〈xn, y〉 − 〈x, y〉‖ add and subtract ‖〈xn, yn〉 − 〈x, y〉‖ = ‖〈xn, yn − y〉 + 〈xn − x, y〉‖
applying the triangular inequality |a + b| ≤ |a| + |b|
‖〈xn, yn〉 − 〈x, y〉‖ ≤ ‖〈xn, yn − y〉‖ + ‖〈xn − x, y〉‖
Using the Cauchy Schwarz inequality
|〈x, y〉|^2 ≤ 〈x, x〉〈y, y〉 |〈x, y〉|^2 ≤ ‖x‖^2 ‖y‖^2 ⇒ |〈x, y〉| ≤ ‖x‖‖y‖
GIM/ DWD 2011 17
‖〈xn, yn〉 − 〈x, y〉‖ ≤ ‖〈xn, xn〉〈yn − n, yn − y〉‖ + ‖〈xn, x〉〈y, y〉‖ ‖〈xn, yn〉 − 〈x, y〉‖ ≤ ‖xn‖‖yn − y‖ + ‖xn − x‖‖y‖ ‖〈xn, yn〉 − 〈x, y〉‖ ≤ ‖xn‖(0) + (0)‖y‖ ‖〈xn, yn〉 − 〈x, y〉‖ ≤ 0 I
From property (i) of norms
‖〈xn, yn〉 − 〈x, y〉‖ ≥ 0 II
So ‖〈xn, yn〉 − 〈x, y〉‖ → 0 as n → ∞, thus continuous.
Note 3.7.1. ‖ <, > ‖ = ‖F‖ = |F| the absolute of a scalar.
3.8 Examples of Hilbert Spaces
A Hilbert space (H) is a vectorspace X over a field F together with an inner product 〈·, ·〉.
- The space X = Rn^ together with inner product
〈x, y〉 = x 1 y 1 + x 2 y 2 + · · · + xnyn (3.6)
is a Hilbert space, where x = (x 1 , y 2 , · · · , xn) and y = (y 1 , y 2 , · · · , yn) and
‖x‖ = 〈x, x〉
(^12)
x^21 + x^22 + · · · + x^2 n
and
d(x, y) = ‖x − y‖ = 〈x − y, x − y〉
(^12)
(x 1 − y 1 )^2 + (x 2 − y 2 )^2 + · · · + (xn − yn)^2
- The space Cn^ is defined by
Cn^ = {(z 1 , z 2 , · · · , zn) : zi ∈ C, i = 1, 2 , · · · , n}
together with inner product
〈z, w〉 = z 1 w 1 + z 2 w 2 + · · · + znwn (3.7)
is a Hilbert space.
GIM/ DWD 2011 19
For example, if f(t) = t^2 on [0, 1]
‖f‖ = max t∈[0,1]
|f(t)| = max t∈[0,1]
is not a Hilbert space, because the norm does not satisfy the parallelogram identity, thus the inner product does not satisfy the parallelogram identity.
2 ‖x‖^2 + 2‖y‖^2 = ‖x + y‖^2 + ‖x − y‖^2
Note 3.9.1. Not all normed vector spaces are Hilbert spaces.
Definition 3.9.1. Complete normed vector spaces are called Banach spaces.
Note 3.9.2. All Hilbert spaces are Banach spaces. The vector space of all p-summable infinite sequences. It’s a Banach space, i.e it’s complete and the norm is given by
‖x‖p =
j=
|ξj |p
) (^1) p (3.11)
Note 3.9.3. With the definition
1 2
It means that given 〈·, ·〉 can be used to generate ‖ · ‖ but the converse is not true.
Note 3.9.4. lp^ is not a Hilbert space for p 6 = 2 and hence not a Hilbert space (This is a Banach space but not a Hilbert space except for p = 2).
Proof. The norm of lp^ with p 6 = 2 cannot be obtained from an inner product since it does not satisfy the parallelogram identity.
2 ‖x‖^2 + 2‖y‖^2 = ‖x + y‖^2 + ‖x − y‖^2
For example, let
x = (1, 1 , 0 , 0 , 0 , · · · ) ∈ lp y = (1, − 1 , 0 , 0 , 0 , · · · ) ∈ lp
20 Functional analysis lecture Notes
‖x‖ = [| 1 |p^ + | 1 |p^ + | 0 |p^ + | 0 |p^ + · · · ]
1 p
1 p ‖y‖ = [| 1 |p^ + | − 1 |p^ + | 0 |p^ + | 0 |p^ + · · · ]
(^1) p
(^1) p
‖x + y‖ = [| 2 |p^ + | 0 |p^ + | 0 |p^ + | 0 |p^ + · · · ]
(^1) p
= 2 ‖x − y‖ = [| 0 |p^ + | 2 |p^ + | 0 |p^ + | 0 |p^ + · · · ]
(^1) p
= 2
2 ‖x‖^2 + 2‖y‖^2 = ‖x + y‖^2 + ‖x − y‖^2
(^1) p^ )^2
(^1) p^ )^2 6 = 4 + 4
(^2) p^ ) = 8
Which is only true when p = 2. lp^ Does not satisfy the parallegram identity for p 6 = 2, thus the norm doesnot induce an inner product, so not a Hilbert space.
Theorem 3.9.1. Completeness: Relative to the induced metric d(x, y) = ‖x − y‖, the Hilbert space H is complete (A metric space is complete if all the Cauchy sequences converge).
Example 3.9.1. The Euclidean space is Complete. Completeness of Rn
Consider a Cauchy sequence (xm) (= xm but since we will have x 1 , x 2 , · · · , xn we prefer to put it up)
Writing
xm^ = (xm 1 , xm 2 , · · · , xmn )
