11 - Inner Product Spaces, Symmetric Operators, Orthogonality

11 - Inner Product Spaces, Symmetric Operators, Orthogonality

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Unformatted text preview: MATH 220: INNER PRODUCT SPACES, SYMMETRIC OPERATORS, ORTHOGONALITY When discussing separation of variables, we noted that at the last step we need to express the inhomogeneous initial or boundary data as a superposition of functions arising in the process of separation of variables. For instance, for the Dirichlet problem Δ u = 0, u | ∂D = h , we had to express h as (1) h ( θ ) = A + ∞ summationdisplay n =1 R n ( A n cos( nθ ) + B n sin( nθ )) , i.e. we had to find constants A n and B n so that this expression holds. The basic framework for this is inner product spaces, which we now discuss. Definition 1. An inner product on a complex vector space V is a map ( .,. ) : V × V → C such that (i) ( .,. ) is linear in the first slot: ( c 1 v 1 + c 2 v 2 ,w ) = c 1 ( v 1 ,w ) + c 2 ( v 2 ,w ) , c 1 ,c 2 ∈ C , v 1 ,v 2 ,w ∈ V, (ii) ( .,. ) is Hermitian symmetric: ( v,w ) = ( w,v ) , with the bar denoting complex conjugate, (iii) ( .,. ) is positive definite: v ∈ V ⇒ ( v,v ) ≥ , and ( v,v ) = 0 ⇔ v = 0 . A vector space with an inner product is also called an inner product space. While one should write ( V, ( ..,. ) ) to specify the inner product space, one typically says merely that V is an inner product space when the inner product is understood. For real vector spaces, one makes essentially the same definition, except that, as the complex conjugate does not make sense, one simply has symmetry: V real vector space ⇒ ( v,w ) = ( w,v ) , v,w ∈ V. We also introduce the notation for the norm associated to this inner product: bardbl v bardbl = ( v,v ) 1 / 2 , where the square root is the unique non-negative square root of a non-negative number (see (iii)). Thus, ( v,v ) = bardbl v bardbl 2 . We note some examples: (i) V = R n , with inner product ( x,y ) = n summationdisplay j =1 x j y j , where x = ( x 1 ,...,x n ), y = ( y 1 ,...,y n ). Thus bardbl x bardbl 2 = ∑ n j =1 x 2 j . 1 2 (ii) V = C n , with inner product ( x,y ) = n summationdisplay j =1 x j y j , where x = ( x 1 ,...,x n ), y = ( y 1 ,...,y n ). Thus bardbl x bardbl 2 = ∑ n j =1 | x j | 2 , which explains why we need Hermitian symmetry for complex vector spaces. (iii) V = R n , with inner product ( x,y ) = n summationdisplay j =1 a j x j y j , where x = ( x 1 ,...,x n ), y = ( y 1 ,...,y n ), a j > 0 for all j . Thus bardbl x bardbl 2 = ∑ n j =1 a j x 2 j . (iv) V = C ( Ω) (complex valued continuous functions on the closure of Ω), where Ω is a bounded domain in R n , with inner product ( f,g ) = integraldisplay Ω f ( x ) g ( x ) dx. Thus, bardbl f bardbl 2 = integraldisplay Ω | f ( x ) | 2 dx. We often write bardbl f bardbl L 2 = bardbl f bardbl L 2 (Ω) for this norm....
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11 - Inner Product Spaces, Symmetric Operators, Orthogonality

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