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Chapter 7 : Existence Theorems 7.2 : A Hilbert Space Approach Although the discussion focuses on the Dirichlet problem , the idea are widely applicable to the study of PDE in general u 0 for u f for 2 Vector Space S = set of objects , closed under addition and scalar multiplication for u, v S, a, b R Example then au bv S x S R 2 : x, y R y the vector space of real - valued S C ([ a, b]) continuous functions defined on [a, b] Chapter 7 : Existence Theorems 7.2 : A Hilbert Space Approach Inner Product This operation associates with each pair of objects x and y In the vector space, a real number denoted <x,y>. The inner product is assumed to satisfy: 1) commutativ e : x,y y,x 2) linear : x y, z x,z y,z 3) For any nonzero vector x, x,x is positive 4) if 0 denotes the zero vector : Example x x In R 2 , x, y 1 , 2 y1 y2 0,0 0 x1 x2 y1 y2 b In, C ([a, b]) f,g f(x)g(x)dx a f x, g 1 x.a 0, b 1 Chapter 7 : Existence Theorems 7.2 : A Hilbert Space Approach Orthogonal: f and g are orthogonal if f,g 0 Example In R 2 , x x x, y 1 , 2 y1 y2 b In, C ([a, b]) f,g f(x)g(x)dx a x1x2 y1 y2 Chapter 7 : Existence Theorems 7.2 : A Hilbert Space Approach Norm The norm of a vector x is a real number x following properties: with the 1) x 0 for all x S 2) x 0 if and only if x is the zero vector. 3) cx c x for any vector x and scalar c. 4) x y x y Example In R 2 , In, C ([ a, b]) x x12 x22 1 2 b f [ f(x)]2 dx a In, C () 1 2 f [ f(x)]2 Chapter 7 : Existence Theorems 7.2 : A Hilbert Space Approach Cauchy-Schwarz Inequality u,v u v Example In R 2 , In, C ([ a, b]) x x12 x22 1 2 b f [ f(x)]2 dx a In, C () 1 2 f [ f(x)]2 Chapter 7 : Existence Theorems 7.2 : A Hilbert Space Approach Length of a vector 2 In R , x x ( a, b) x12 x x22 Distance between 2 vectors x y x y (a1 a2 ) 2 (b1 b2 ) 2 y (a2 , b2 ) x (a1 , b1 ) Chapter 7 : Existence Theorems 7.2 : A Hilbert Space Approach Norm induced by an inner product x x, x 2 In R , In R 2 , x x, x x12 x22 f f,f In C ([a, b]), b f ( x) dx a 2 Chapter 7 : Existence Theorems 7.2 : A Hilbert Space Approach Convergent sequence v1 , v2 , v3 ,V converges to v V If We say the sequence : lim vn v 0 n Example In R 2 , x x12 x22 In, C ([ a, b]) 1 2 f [ f(x)]2 dx a b Chapter 7 : Existence Theorems 7.2 : A Hilbert Space Approach Cauchy sequence We say the sequence : v1, v2 , v3 ,V is cauchy If lim vn vm 0 n ,m Example: In R 2 , x x12 x22 In, C ([ a, b]) 1 2 f [ f(x)]2 dx a b Remark: Every convergent sequence in V is cauchy. (proof) Chapter 7 : Existence Theorems 7.2 : A Hilbert Space Approach Complete Space A normed space is complete if every cauchy sequence in V is convergent. Hilbert Space A compete inner product space ( with respect the norm induced by the inner product) is called a Hilbert Space. Chapter 7 : Existence Theorems 7.2 : A Hilbert Space Approach Hilbert Space A compete inner product space ( with respect the norm induced by the inner product) is called a Hilbert Space. Example R n is a Hilbert space, because any cauchy sequence of n - vectors converges to a vector in R n . C ([0,1]) ??? Hilbert space ??? Chapter 7 : Existence Theorems 7.2 : A Hilbert Space Approach Hilbert Space A compete inner product space ( with respect the norm induced by the inner product) is called a Hilbert Space. Example L2 ([0,1]) the space of real - valued functions defined 1 on the interval [0,1] and such that f 2 0 1 Is a Hilbert space with the inner product . f , g fg 0 and norm 1 2 1 2 f f 0 Chapter 7 : Existence Theorems 7.2 : A Hilbert Space Approach Hilbert Space A compete inner product space ( with respect the norm induced by the inner product) is called a Hilbert Space. Example compact set on R 2 L2 () the space of real - valued functions defined on and such that f 2 Is a Hilbert space with the inner product . f , g fg and norm f f 1 2 2 Chapter 7 : Existence Theorems 7.2 : A Hilbert Space Approach Hilbert Space A compete inner product space ( with respect the norm induced by the inner product) is called a Hilbert Space. Example compact set on R 2 L2 () the space of real - valued functions defined on and such that f 2 f , g fg Chapter 7 : Existence Theorems 7.2 : A Hilbert Space Approach The completion of V If V is an inner product space that is not complete, then V can be embedded as a dense subset of a Hilbert space H. This means that there is a Hilbert space such that V inside H. We call such H the completion of V. Example x Q 2 : x, y are rational numbers y It is an inner product space with standard inner product. It is not complete ( give an example) The completion of Q2 is R2 Chapter 7 : Existence Theorems 7.2 : A Hilbert Space Approach The completion of V If V is an inner product space that is not complete, then V can be embedded as a dense subset of a Hilbert space H. This means that there is a Hilbert space such that V inside H. We call such H the completion of V. Example consisting all functions that are cont., f , g fg f x g x f y g y C () with cont. first partial derivative s 1 C01 () f : f C1 (), f 0 on f , g fg f x g x f y g y It is an inner product space. But It is not complete The completion of H 01 () f : C01 () is H 01 () f f 2 f , , L (), and f 0 on x y Chapter 7 : Existence Theorems 7.2 : A Hilbert Space Approach Linear functional A linear functional on an inner product space V is a realvalued function : V R satisfying the linearity condition (x y) ( x) ( y) 1 In L ([0,1]) , ( f ) 0 f ( x) sin( x)dx 2 Example Bounded Linear functional A linear functional is bounded if there exist a positive M, such that ( x) M x Example x V 1 In L2 ([0,1]) , ( f ) 0 f ( x) sin( x)dx Chapter 7 : Existence Theorems 7.2 : A Hilbert Space Approach Bounded Linear functional and inner product Bounded linear functionals on an inner product space V are intimately tied to the inner product on V. if y0 V , then ( x) x, y0 ? define a bounded linear functional in V Is the other way true??? Riesz Representation Theorem: If is any bounded linear functional on a Hilbert space H, then there is a unique y0 H such that ( x) x, y0 Riesz Representation Theorem: Every bounded linear functional on a Hilbert space can be written as an inner product with some fixed vector in the space. Chapter 7 : Existence Theorems 7.2 : A Hilbert Space Approach Bilinear Form a function or functional of two variables that is linear with respect to each variable when the other variable is held fixed. 1) a(u,v1 v2 ) a(u, v1 ) a(u, v2 ) 2) a(u1 u2 , v) a(u1, v) a(u2 , v) Example Chapter 7 : Existence Theorems 7.2 : A Hilbert Space Approach Bounded Form or continuous A bilinear form a(.,.) on a Hilbert space H is said to be continouos, if there exists a positive constant C such that a(u, v) C u H v H u, v H Coercive Form A bilinear form a(.,.) on a Hilbert space H is said to be coercive, if there exists a positive constant such that a(u, u ) u 2 H u H Chapter 7 : Existence Theorems 7.2 : A Hilbert Space Approach Consider the following problem Find u H such that a(u, v) F (v) v H Variational Problem Where H is a Hilbert space with (. , .) inner product F () is a bounded linear functional on H ( F H' ) a(,) bounded, coercive, bilinear form We want to study this problem (existence and uniquence) Chapter 7 : Existence Theorems 7.2 : A Hilbert Space Approach The Lax-Milgram Theorem Given: a Hilbert space H,(.,.), a continuous, coercive bilinear form a(.,.) and a continuous linear functional F in H’, There exists a unique u in H such that a(u, v) F (v) v H Chapter 7 : Existence Theorems 7.2 : A Hilbert Space Approach Example 1) Bilinear form 2) Bounded (cont.) 3) Coercive