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N. Let x be the n-tuple x = (x1 , . . , xn ). Then x is just a point in the n-dimensional Euclidean space, x ∈ Rnx . Let us introduce the dual space Rnp whose points are n-tuples p = (p1 , . . , pn ). 65) Rn where dx = dx1 . . dxn , and p, x = form is then defined by n j=1 f (x) = (F −1 fˆ)(x) = Rn pj xj . The inverse Fourier trans- dp i e (2π)n p,x fˆ(p) . 66) 14 1 Background in Analysis The n-dimensional delta-function can be represented as the Fourier integral δ(x) = Rn dp i e (2π)n p,x .

1 First Order Partial Differential Equations Partial differential equations are usually much more complicated than ordinary differential equations, even systems of ordinary differential equations. However, there is one class of partial differential equations that is as difficult to solve as a system of ordinary differential equations. These are linear first-order partial differential equations. Let xi , i = 1, . . , n, be the coordinates in the Euclidean space Rn and i ξ (x) be a vector field in Rn . Let f be a given smooth function in Rn and L be a first-order partial differential operator of the form n ξ i (x) L= i=1 ∂ .

Similarly, let M be an open set in Rn (in particular, M can be the whole n R ) and μ be a positive function on M . Then the inner product can be defined by dx μ(x) f (x)h(x) . 131) M This defines the Hilbert space L2 (M, μ) of complex valued functions with finite L2 -norm. The spaces of smooth functions are not complete and, therefore, are not Hilbert spaces. This is a simple consequence of the fact that there are convergent sequences of smooth functions whose limit is not a smooth function. The L2 spaces of square integrable functions are obtained by adding the limits of all convergent sequences to the spaces of smooth functions.

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