6 1. Vecto r Spaces d. Le t X b e a finite set. C(X) denote s the set of all complex-valued functions o n X, with the standard addition of functions, and multi- plication of functions by scalars. e. Th e se t C R ([0,1]) o f al l continuou s real-value d function s / o n [0,1], and the set C([0,1]) of all continuous complex-valued func - tions / o n [0,1] , with the standard operations of addition of func- tions and of multiplication o f functions b y scalars. C R ([0,1]) i s a real vector space. C([0,1] ) i s naturally a complex vector space, but becomes a real vector space if we limit the allowable scalars to real numbers only. /. Th e set C°°([— 1,1]) o f all infinitely differentiate real-value d func- tions / on [—1,1] , with the standard operations on functions i s a real vector space. g. Th e se t 3T N o f 27T-periodi c trigonometri c polynomial s o f degre e bounded by N that is, the functions of the form E|rtiA^^^mx. Th e underlying field may be C or R, and the operations are the standard addition of functions an d multiplication of functions by scalars. Similarly, th e spac e ^ NM o f trigonometri c polynomial s i n tw o variables o f the form Y,\n\N \ m \Mel^nx+my^ w ^ ^ e s a m e °P er_ ations and the same underlying field(s). h. Th e set of complex-valued functions / o n R that satisfy the differ- ential equation 3f'(x)-smxf{x) + 2f(x)=0, with the standard operations on functions. I f we are interested in real-valued functions only , the underlying field is naturally R. I f we allow complex-valued functions w e may choose either C or R as the underlying field. 1.2.2 Isomorphism . Th e expression "virtually identical" in the com- parison abov e of ./#(« , m F) wit h ¥ mn i s not a proper mathematica l term. The proper term here is isomorphic.

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