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Unformatted text preview: 642:527 FALL 2009 EXPANSIONS IN ORTHOGONAL BASES Vector spaces We will use without much further comment the idea of a vector space . Basically, a vector space is a set of vectors (these vectors will in fact often be functions) with the property that a linear combination of vectors is again a vector. Linear combinations involve scalars , which may be real numbers (for a real vector space ) or complex numbers (for a complex vector space ). If u and v are vectors in a vector space, and α and β are scalars, then αf + βg is also a vector in the space. These linear combinations are required to satisfy a set of rules which any reasonable person would consider obvious. See Section 9.6 of Greenberg, and in particular Definition 9.6.1, for a careful discussion of vector spaces. For the moment we will consider only real vector spaces, returning to complex vector spaces at the end of these notes. Example 1: One familiar vector space is R n , the set of all row vectors with n (real) components. If u = ( u 1 , . . ., u n ) and v = ( v 1 , . . ., v n ) are two vectors in R n then their linear combinations are constructed by making linear combinations of their components: α u + β v = ( αu 1 + βv 1 , . . ., αu n + βv n ) . (1) Example 2: Another example of a vector space, important for the theory of Fourier series and similar applications, is C p [ a, b ], the set of all piecewise continuous, real-valued functions f ( x ) defined for a ≤ x ≤ b . (Piecewise continuity is defined on page 249 of Greenberg.) As in any space of functions, the rule for linear combinations is ( αf + βg )( x ) = αf ( x ) + βg ( x ) , a ≤ x ≤ b. (2) An element f of C p [ a, b ] is of course a function but, since it belongs to a vector space, we may speak of it as a “vector” when we want to emphasize this context. Read Section 17.6 of Greenberg for more on C p [ a, b ] as a vector space. Notation: When we speak of a general vector space in these notes we will denote typical vectors as in Example 1, using boldface letters: u , v , etc., and later e 1 , e 2 , . . . . The reader should bear in mind, however, that what we say applies equally well when the vectors under consideration are functions, considered as members of a vector space like C p [ a, b ]. When we are speaking specifically about functions we will denote them, as in Example 2, by the letters f , g , etc. 1 640:527 EXPANSIONS IN ORTHOGONAL BASES FALL 2009 Inner products An inner product in a vector space is a formula which assigns to any pair of vectors, say u and v , a number ( u , v ) , their inner product. When the vector space is R n there is a familiar inner product (usually called the dot product): ( u , v ) = u · v = n summationdisplay i =1 u i v i . (3) When the vector space is C p [ a, b ] the most common inner product, used in the study of Fourier series, is ( f, g ) = integraldisplay b a f ( x ) g ( x ) dx. (4) Inner products are discussed further in Section 9.6.2 of Greenberg....
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