Advanced Computational Fluid Dynamics AA215A Lecture 1 Vector and Function Spaces Antony Jameson Winter Quarter 2013 Stanford CA Last revised on February 5 2013 Contents 1 Vector and Function Spaces 1 1 Basic concepts 1 2 Convex and concave functions Jensen s theorem 1 3 The generalized AM GM inequality 1 4 Ho lder s inequality and the Cauchy Schwarz inequality 1 5 Vector Norms 1 5 1 Minkowski s inequality triangle inequality for the p norm 1 5 2 Equivalence of Norms for Finite Dimensional Vectors 1 6 Matrix Norms 1 6 1 Infinity Norm 1 6 2 1 Norm 1 6 3 2 Norm 1 7 Norms of Functions 1 7 1 Function Space 1 7 2 Norms 1 7 3 Non equivalence of Norms for Functions 1 8 Inner Products of Functions Orthogonality 1 8 1 Orthogonality 1 8 2 Triangle Inequality for the and 1 Norms 1 8 3 Cauchy Schwarz Inequality 1 8 4 Triangle Inequality for the Euclidean Norm 1 8 5 Pythagorean Theorem 1 8 6 Linear Independence 1 8 7 Weierstrass Theorem 1 9 1 2 2 2 3 3 4 5 5 6 7 7 7 7 7 8 9 10 10 10 10 11 11 11 11 11 Lecture 1 Vector and Function Spaces 1 1 Basic concepts The well known concepts of Euclidean space can be generalized to n dimensional vectors and also to functions These concepts are extremely useful in the analysis of numerical methods and are briefly reviewed in this appendix The idea of function space was one of the great achievements of nineteenth century mathematics particularly due to Banach and Hilbert Hilbert was the first to introduce spaces with abstract inner product As a preliminary it is useful to derive some basic inequalities due to Jensen Ho lder Cauchy and Schwarz 1 2 Convex and concave functions Jensen s theorem A function of f x of a real variable x is convex if f tx1 1 t x2 tf x1 1 t f x2 1 1 f tx1 1 t x2 tf x1 1 t f x2 1 2 and concave if whenever 0 t 1 It is strictly convex or concave if equality in these expressions implies x1 x2 If f is twice differentiable it is convex if f 00 0 and concave if f 00 0 and strictly convex or concave if f 00 0 or f 00 0 Jensen s theorem If f x is a concave function n n X X ti f xi f ti xi 1 3 Pn i 1 i 1 whenever t1 tn 0 1 and i 1 ti 1 This may be proved by induction It is true for n 2 by the definition of a concave function Suppose n 3 and the assertion holds for smaller values of n Then for i 2 n set t0i ti 1 t1 2 LECTURE 1 VECTOR AND FUNCTION SPACES so that Pn 0 i 2 ti 3 1 Then n X ti f x1 t1 f x1 1 t1 i 1 n X t0i f xi i 2 t1 f x1 1 t1 f f t1 x1 1 f n X n X t0i xi i 2 n X 0 t1 ti x i i 2 ti xi i 1 1 3 The generalized AM GM inequality The geometric mean GM of n real numbers does not exceed the arithmetic mean AM This result follows from Jensen s theorem applied to the function log x which is strictly concave Thus P if p1 pn 0 and ni 1 pi 1 n n X X pi log ai log pi ai 1 4 i 1 i 1 and hence n Y api i i 1 n X pi a i 1 5 i 1 This is known as the generalized AM GM inequality Setting pi inequality n 1 n n Y 1X ai ai n i 1 1 4 1 n recovers the basic AM GM 1 6 i 1 Ho lder s inequality and the Cauchy Schwarz inequality Suppose p q 1 and 1 p 1 q 1 Then 1 n 1 n n p q X X X p q ak ak bk bk k 1 k 1 1 7 k 1 Set x1 ap x2 bq p1 p1 p2 1q By the generalized GM AM theorem ab xp11 xp22 p1 x1 p2 x2 Take vectors such that n X k 1 ak p n X k 1 ap bq p q bk q 1 1 8 1 9 LECTURE 1 VECTOR AND FUNCTION SPACES Then 4 n n n X X X ak p bk q 1 1 a b a b 1 k k k k p q p q k 1 k 1 1 10 k 1 When p 2 this reduces to the Cauchy Schwartz inequality n n 1 n 1 2 2 X X X ak bk a2k b2k k 1 k 1 1 11 k 1 This may be proved directly by noting that for any value of n X ak bk ak bk 0 1 12 k 1 and consequently n X k 1 n n X X 2 2 ak 2 ak bk b2k 0 k 1 Then setting and multiplying by Pn 2 k 1 bk P nk 1 ak bk Pn 2 k 1 bk n X a2k k 1 1 5 n X k 1 1 13 k 1 b2k n X ak bk 1 14 1 15 k 1 Vector Norms The size of an n dimensional vector can be conveniently represented by a variety of measures Such measures are called norms which are required to satisfy the following axioms To each vector x assign a number x where 1 x 0 2 x x for scalar 3 x y x y triangle inequality 4 x 0 if and only if x 0 Some widely used norms are 1 P x p xi p p for p 1 2 3 P x 1 xi x 2 P xi 2 1 x max xi i 2 LECTURE 1 VECTOR AND FUNCTION SPACES 5 Conditions 1 and 2 are evident For 3 P P x y 1 xi yi xi yi x 1 y 1 x y max xi yi max xi max yi x y i i i To verify 3 for x 2 use Cauchy Schwartz inequality xT y x y 1 16 Then 1 x y 2 x y T x y 2 1 T 12 T 1 1 2 T T T T T 2 2 x x 2 y x y y x x 2 x x y y y y x 2 y 2 1 5 1 Minkowski s inequality triangle inequality for the p norm x y p x p y p 1 17 Proof X xk yk p X xk yk p 1 xk X xk yk p 1 yk 1 18 Then by Ho lder s inequality X p xk yk X X p 1 q xk yk p xk yk 1 X q 1 X q 1 xk p xk p 1 p p X X p yk …