We assign a complex number Am to each n-tuple of non-negative integers m = (m₁, ..., mn) arbitrarily. Show that there exists an f(x1,...,xn) E C (R") satisfying D"f(0) = Am for any m, where 0 = (0,..., 0). The one variable case was shown by Borel (1895). Later Rosenthal (1953) gave a simpler proof by considering 8(x)=ane-lanin!x2 xh n=0 is determined according to the given value of g)(0). Mirkil (1956) gave a proof for the n-dimensional case.

We assign a complex number Am to each n-tuple of non-negative integers m = (m₁, ..., mn) arbitrarily. Show that there exists an f(x1,...,xn) E C (R") satisfying D"f(0) = Am for any m, where 0 = (0,..., 0). The one variable case was shown by Borel (1895). Later Rosenthal (1953) gave a simpler proof by considering 8(x)=ane-lanin!x2 xh n=0 is determined according to the given value of g)(0). Mirkil (1956) gave a proof for the n-dimensional case.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter6: Vector Spaces

Section6.2: Linear Independence, Basis, And Dimension

Problem 61EQ

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