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Unformatted text preview: Q ( v ) = 2 Q ( v ) for each R and v V . (b) Show how the polarization identity provides a means of dening a symmetric bilinear map given a quadratic function on a vector space. 5. Let [ a,b ] be an interval and let L 2 ([ a,b ]) be the set of functions f : [ a,b ] R whose square is integrable over [ a,b ]. We refer to such functions as square-integrable . Show that the map ( f,g ) 7 R b a f ( t ) g ( t ) dt is an inner product. This map is referred to as the L 2 ([ a,b ])-inner product; it is convenient to denote it by f,g L 2 ([ a,b ]) . 6. For V a nite-dimensional R-vector space, show that the following pairs of vector spaces are naturally isomorphic by providing explicit isomorphisms: (a) T 1 ( V ) and V ; (b) T 1 ( V ) and V * ; (c) T 1 1 ( V ) and L ( V ; V ); (d) T 2 ( V ) and L ( V ; V * ); (e) T 2 ( V ) and L ( V * ; V ). 1...
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- Spring '10