Appendix. Banach and Hilbert Spaces. The Lp-Spaces
In this appendix we shall first recall some basic concepts from calculus without proofs. After that, we shall prove some smoothing results for Lp-functions. Definition A.1. A Banach space B is a real vector space that is equipped with a norm k k that satisfies the following properties: (i) kxk >0for all x 2 B, x ¤ 0. (ii) k˛xk D j˛j kxk for all ˛ 2 R, x 2 B. (iii) kx C yk kxk C kyk for all x;y 2 B (triangle inequality). (iv) B is complete with respect to k k (i.e., every Cauchy sequence has a limit in B). We recall the Banach fixed-point theorem Theorem A.1. Let .B; k k/ be a Banach space, A B a closed subset, and f W A ! B a map with f.A/ A which satisfies the inequality
kf.x/ f.y/k kx yk for all x;y 2 A; for some fixed with 0 <1: Then f has unique fixed point in A, i.e., a solution of f.x/ D x. For example, every Hilbert space is a Banach space. We also recall that concept: Definition A.2. A (real) Hilbert space H is a vector space over R, equipped with a scalar product
. ; / W H H ! R that satisfies the following properties: (i) .x; y/ D .y; x/ for all x;y 2 H. (ii) . 1x1 C 2x2;y/D 1.x1;y/C 2.x2;y/for all 1; 2 2 R, x1;x2;y 2 H. (iii) .x; x/ > 0 for all x ¤ 0, x 2 H.
J. Jost, Partial Differential Equations, Graduate Texts in Mathematics 214, 393 DOI 10.1007/978-1-4614-4809-9, © Springer Science+Business Media New York 2013 394 Appendix. Banach and Hilbert Spaces. The Lp-Spaces
(iv) H is complete with respect to the norm
1 kxk WD .x; x/ 2 :
In a Hilbert space H, the following inequalities hold: – Schwarz inequality:
j.x; y/j kxk kyk ; (A.1)
with equality precisely if x and y are linearly dependent. – Triangle inequality:
kx C yk kxk C kyk : (A.2)
Likewise without proof, we state the Riesz representation theorem: Let L be a bounded linear functional on the Hilbert space H, i.e., L W H ! R is linear with
jLxj kLk WD sup < 1: x¤0 kxk
Then there exists a unique y 2 H with L.x/ D .x; y/ for all x 2 H, and
kLk D kyk :
The following extension is important, too: Theorem of LaxÐMilgram: Let B be a bilinear form on the Hilbert space H that is bounded,
jB.x; y/j K kxkkyk for all x;y 2 H with K<1; and elliptic, or, as this property is also called in the present context, coercive,
jB.x;x/j kxk2 for all x 2 H with >0:
For every bounded linear functional T on H, there then exists a unique y 2 H with
B.x; y/ D Tx for all x 2 H:
Proof. We consider
Lz.x/ D B.x; z/: Appendix. Banach and Hilbert Spaces. The Lp -Spaces 395
By the Riesz representation theorem, there exists Sz 2 H with
.x; Sz/ D Lzx D B.x; z/:
Since B is bilinear, Sz depends linearly on z. Moreover,
kSzk K kzk :
Thus, S is a bounded linear operator. Because of