Arnold Douglas - Functional Analysis.pdf - Analysis - Kuya

FUNCTIONAL ANALYSIS 1

Douglas N. Arnold 2

References:

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.

CONTENTS

I. Vector spaces and their topology:::::::::::::::::::::::::::::::::::::::::::::::2

Subspaces and quotient spaces :::::::::::::::::::::::::::::::::::::::::::: 4

Basic properties of Hilbert spaces ::::::::::::::::::::::::::::::::::::::::: 5

II. Linear Operators and Functionals::::::::::::::::::::::::::::::::::::::::::::::9

The Hahn{Banach Theorem::::::::::::::::::::::::::::::::::::::::::::::10

Duality :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 10

III. Fundamental Theorems:::::::::::::::::::::::::::::::::::::::::::::::::::::::14

The Open Mapping Theorem:::::::::::::::::::::::::::::::::::::::::::::14

The Uniform Boundedness Principle::::::::::::::::::::::::::::::::::::::15

The Closed Range Theorem :::::::::::::::::::::::::::::::::::::::::::::: 16

IV. Weak Topologies ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 18

The weak topology:::::::::::::::::::::::::::::::::::::::::::::::::::::::18

The weak* topology::::::::::::::::::::::::::::::::::::::::::::::::::::::19

V. Compact Operators and their Spectra :::::::::::::::::::::::::::::::::::::::: 22

Hilbert{Schmidt operators ::::::::::::::::::::::::::::::::::::::::::::::: 22

Compact operators:::::::::::::::::::::::::::::::::::::::::::::::::::::::23

Spectral Theorem for compact self-adjoint operators :::::::::::::::::::::: 26

The spectrum of a general compact operator ::::::::::::::::::::::::::::: 28

VI. Introduction to General Spectral Theory::::::::::::::::::::::::::::::::::::::31

The spectrum and resolvent in a Banach algebra ::::::::::::::::::::::::: 31

Spectral Theorem for bounded self-adjoint operators::::::::::::::::::::::35

1 These lecture notes were prepared for the instructor's personal use in teaching a half-semester course

on functional analysis at the beginning graduate level at Penn State, in Spring 1997. They are certainly

not meant to replace a good text on the subject, such as those listed on this page.

2 Department of Mathematics, Penn State University, University Park, PA 16802.

Email: dna@math.psu.edu. Web: http://www.math.psu.edu/dna/.

I. Vector spaces and their topology

Basic denitions: (1) Norm and seminorm on vector spaces (real or complex). A norm

denes a Hausdor topology on a vector space in which the algebraic operations are con-

tinuous, resulting in a normed linear space. If it is complete it is called a Banach space.

(2) Inner product and semi-inner-product. In the real case an inner product is a positive

denite, symmetric bilinear form on XX !R. In the complex case it is positive denite,

Hermitian symmetric, sesquilinear form X X ! C. An (semi) inner product gives rise

to a (semi)norm. An inner product space is thus a special case of a normed linear space.

A complete inner product space is a Hilbert space, a special case of a Banach space.

The polarization identity expresses the norm of an inner product space in terms of the

inner product. For real inner product spaces it is

(x;y) =

4 (kx + yk 2 kxyk 2 ):

For complex spaces it is

(x;y) =

4 (kx + yk 2 + ikx + iyk 2 kxyk 2 ikxiyk 2 ):

In inner product spaces we also have the parallelogram law:

kx + yk 2 + kxyk 2 = 2(kxk 2 + kyk 2 ):

This gives a criterion for a normed space to be an inner product space. Any norm coming

from an inner product satises the parallelogram law and, conversely, if a norm satises the

parallelogram law, we can show (but not so easily) that the polarization identity denes

an inner product, which gives rise to the norm.

(3) A topological vector space is a vector space endowed with a Hausdor topology such

that the algebraic operations are continuous. Note that we can extend the notion of Cauchy

sequence, and therefore of completeness, to a TVS: a sequence x n in a TVS is Cauchy if

for every neighborhood U of 0 there exists N such that x m x n 2 U for all m;n N.

A normed linear space is a TVS, but there is another, more general operation involving

norms which endows a vector space with a topology. Let X be a vector space and suppose

that a family fkk g 2A of seminorms on X is given which are sucient in the sense that

d(x;y) =

2 n kxyk n

fkxk = 0g = 0. Then the topology generated by the sets fkxk < rg, 2A, r > 0,

makes X a TVS. A sequence (or net) x n converges to x i kx n xk ! 0 for all . Note

that, a fortiori, jkx n k kxk j! 0, showing that each seminorm is continuous.

If the number of seminorms is nite, we may add them to get a norm generating the

same topology. If the number is countable, we may dene a metric

1 + kxyk n ;

so the topology is metrizable.

Examples: (0) On R n or C n we may put the l p norm, 1 p 1, or the weighted

l p norm with some arbitrary positive weight. All of these norms are equivalent (indeed

all norms on a nite dimensional space are equivalent), and generate the same Banach

topology. Only for p = 2 is it a Hilbert space.

(2) If is a subset of R n (or, more generally, any Hausdor space) we may dene the

space C b () of bounded continuous functions with the supremum norm. It is a Banach

space. If X is compact this is simply the space C() of continuous functions on .

(3) For simplicity, consider the unit interval, and dene C n ([0; 1]) and C n; ([0; 1]),

n 2 N, 2 (0; 1]. Both are Banach spaces with the natural norms. C 0;1

is the space of

Lipschitz functions. C([0; 1]) C 0; C 0; C 1 ([0; 1]) if 0 < 1.

(4) For 1 p < 1 and an open or closed subspace of R n (or, more generally, a -nite

measure space), we have the space L p () of equivalence classes of measurable p-th power

integrable functions (with equivalence being equality o a set of measure zero), and for

p = 1 equivalence classes of essentially bounded functions (bounded after modication

on a set of measure zero). For 1 < p < 1 the triangle inequality is not obvious, it is

Minkowski's inequality. Since we modded out the functions with L p -seminorm zero, this

is a normed linear space, and the Riesz-Fischer theorem asserts that it is a Banach space.

L 2

is a Hilbert space. If meas() < 1, then L p () L q () if 1 q p 1.

(5) The sequence space l p , 1 p 1 is an example of (4) in the case where the

measure space is N with the counting measure. Each is a Banach space. l 2 is a Hilbert

space. l p l q if 1 p q 1 (note the inequality is reversed from the previous example).

The subspace c 0 of sequences tending to 0 is a closed subspace of l 1 .

(6) If is an open set in R n (or any Hausdor space), we can equip C() with the

norms f 7!jf(x)j indexed by x 2 . This makes it a TVS, with the topology being that

of pointwise convergence. It is not complete (pointwise limit of continuous functions may

not be continuous).

(7) If is an open set in R n we can equip C() with the norms f 7!kfk L 1 (K) indexed

by compact subsets of , thus dening the topology of uniform convergence on compact

subsets. We get the same toplogy by using only the countably many compact sets

K n = fx 2 : jxj n; dist(x;@) 1=ng:

The topology is complete.

(8) In the previous example, in the case is a region in C, and we take complex-

valued functions, we may consider the subspace H() of holomorbarphic functions. By

Weierstrass's theorem it is a closed subspace, hence itself a complete TVS.

(9) If f;g 2 L 1 (I), I = (0; 1) and

f(x)(x) dx =

g(x) 0 (x) dx;

for all innitely dierentiable with support contained in I (so is identically zero near

0 and 1), then we say that f is weakly dierentiable and that f 0 = g. We can then dene

the Sobolev space W p (I) = ff 2 L p (I) : f 0 2 L p (I)g, with the norm

1=p

kfk W p (I) =

jf(x)j p dx +

jf 0 (x)j p dx

This is a larger space than C 1 ( I), but still incorporates rst order dierentiability of f.

The case p = 2 is particularly useful, because it allows us to deal with dierentiability

in a Hilbert space context. Sobolev spaces can be extended to measure any degree of

dierentiability (even fractional), and can be dened on arbitrary domains in R n .

Subspaces and quotient spaces.

If X is a vector space and S a subspace, we may dene the vector space X=S of cosets.

If X is normed, we may dene

x2u kxk X , or equivalently kxk X=S = inf

s2S kxsk X :

This is a seminorm, and is a norm i S is closed.

Theorem. If X is a Banach space and S is a closed subspace then S is a Banach space

and X=S is a Banach space.

Sketch. Suppose x n is a sequence of elements of X for which the cosets x n are Cauchy.

We can take a subsequence with kx n x n+1 k X=S 2 n1 , n = 1; 2;::: . Set s 1 = 0, dene

s 2 2 S such that kx 1 (x 2 +s 2 )k X 1=2, dene s 3 2 S such that k(x 2 +s 2 )(x 3 +s 3 )k X

1=4, ::: . Then fx n + s n g is Cauchy in X :::

A converse is true as well (and easily proved).

Theorem. If X is a normed linear space and S is a closed subspace such that S is a

Banach space and X=S is a Banach space, then X is a Banach space.

Finite dimensional subspaces are always closed (they're complete). More generally:

Theorem. If S is a closed subspace of a Banach space and V is a nite dimensional

subspace, then S + V is closed.

Sketch. We easily pass to the case V is one-dimensional and V \S = 0. We then have that

S+V is algebraically a direct sum and it is enough to show that the projections S+V ! S

and S + V ! V are continuous (since then a Cauchy sequence in S + V will lead to a

Cauchy sequence in each of the closed subspaces, and so to a convergent subsequence).

Now the projection : X ! X=S restricts to a 1-1 map on V so an isomorphism of V onto

its image V . Let : V ! V be the continuous inverse. Since (S + V ) V , we may form

the composition j S+V : S + V ! V and it is continuous. But it is just the projection

onto V . The projection onto S is id, so it is also continuous.

kuk X=S = inf

Note. The sum of closed subspaces of a Banach space need not be closed. For a coun-

terexample (in a separable Hilbert space), let S 1 be the vector space of all real sequences

(x n ) n=1 for which x n = 0 if n is odd, and S 2 be the sequences for which x 2n = nx 2n1 ,

n = 1; 2;::: . Clearly X 1 = l 2 \S 1 and X 2 = l 2 \S 2 are closed subspaces of l 2 , the space

of square integrable sequences (they are dened as the intersection of the null spaces of

continuous linear functionals). Obviously every sequence can be written in a unique way

as sum of elements of S 1 and S 2 :

(x 1 ;x 2 ;::: ) = (0;x 2 x 1 ; 0;x 4 2x 3 ; 0;x 6 3x 5 ;::: ) + (x 1 ;x 1 ;x 3 ; 2x 3 ;x 5 ; 3x 5 ;::: ):

If a sequence has all but nitely many terms zero, so do the two summands. Thus all

such sequences belong to X 1 + X 2 , showing that X 1 + X 2 is dense in l 2 . Now consider the

sequence (1; 0; 1=2; 0; 1=3;::: ) 2 l 2 . Its only decomposition as elements of S 1 and S 2 is

(1; 0; 1=2; 0; 1=3; 0;::: ) = (0;1; 0;1; 0;1;::: ) + (1; 1; 1=2; 1; 1=3; 1;::: );

and so it does not belong to X 1 + X 2 . Thus X 1 + X 2 is not closed in l 2 .

Basic properties of Hilbert spaces.

An essential property of Hilbert space is that the distance of a point to a closed convex

set is alway attained.

Projection Theorem. Let X be a Hilbert space, K a closed convex subset, and x 2 X.

Then there exists a unique x 2 K such that

kx xk = inf

y2K kxyk:

Proof. Translating, we may assume that x = 0, and so we must show that there is a unique

element of K of minimal norm. Let d = inf y2K kyk and chose x n 2 K with kx n k ! d.

Then the parallelogram law gives

x n x m

2 kx n k 2 +

2 kx m k 2

x n + x m

2 kx n k 2 +

2 kx m k 2 d 2 ;

where we have used convexity to infer that (x n + x m )=2 2 K. Thus x n is a Cauchy

sequence and so has a limit x, which must belong to K, since K is closed. Since the norm

is continuous, kxk = lim n kx n k = d.

For uniqueness, note that if kxk = kxk = d, then k(x+ x)=2k = d and the parallelogram

law gives

kx xk 2 = 2kxk 2 + 2kxk 2 kx + xk 2 = 2d 2 + 2d 2 4d 2 = 0:

The unique nearest element to x in K is often denoted P K x, and referred to as the

projection of x onto K. It satises P K P K = P K , the denition of a projection. This

terminology is especially used when K is a closed linear subspace of X, in which case P K

is a linear projection operator.

Arnold Douglas - Functional Analysis.pdf

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