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Forms of Isometries Between Function Spaces Kathleen Roberts1 Iowa State University, Ames, IA

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Forms of Isometries Between Function Spaces Kathleen Roberts1 Iowa State University, Ames, IA Forms of Isometries Between Function Spaces Kathleen Roberts1 Iowa State University, Ames, IA Introduction Takeshi's Challenge Results The classic Banach-Stone Theorem establishes a form for surjective, Consider the unit circle in the complex plane In the Banach-Stone proof we referred to, we were able to complex-linear isometries (distance preserving functions) between function relate the functions we sought, and ', to the inverse A = faz + b : a, b 2 , jzj = 1g spaces. Mathematician Takeshi Miura from Niigata University questioned C image of a set of functions, Q such that for w, α 2 T, what could be said about surjective, real-linear isometries after finding a and isometry Q(w, α) = ff 2 A: f (w) = α, kf k = 1g. counter-example that demonstrated the shortcomings of the Banach-Stone T (az + b) = az + b. T Theorem. Through a careful examination of the Banach-Stone Theorem We found that proof, we hope to gain insight into why the theorem does not hold in general T -1[Q(w, 1)] = Q( (w), '(w)). and to find a form for real-linear isometries between function spaces. This along with the complex-linearity of T allowed us to access any of the complex-numbers. However, since we Objectives sought to create such a generalization for a real-linear function, we needed to use two different such sets, one to I We seek to determine why the Banach-Stone Theorem fails to classify allow us to access any real part and one to allow us to forms of isometries for general subspaces. access any imaginary part: We seek to understand the failure of the classification for the I T -1[Q(w, 1)] = Q( (w), ' (w)) counterexample. 1 1 and I We seek to prove a conjecture as to the correct form of such isometries of -1 the unit circle in the complex plane. T [Q(w, i)] = Q( i(w), 'i(w)). Methods Conclusion I Understand problem background and definitions needed to understand the As stated by the original Banach-Stone Theorem, Banach-Stone Theorem. complex-linear isometries T between function spaces take I Begin by examining a proof of the Banach-Stone Theorem and see where this form: the failure occurs when we have only real-linearity instead of T (f ) = T (1) · (f ◦ ). complex-linearity. However, real-linear isometries T between function spaces I Seek to prove the conjecture for forms of real-linear isometries. take one of the following forms: T (f )(z) = Re[ 1(z)f ( 1(z))] + i Im[ 1(z)f ( i(z))] Classic Banach-Stone Result or The Banach-Stone Theorem says if T is a surjective, complex-linear isometry T (f )(z) = Re[ 1(z)f ( 1(z))] - i Im[ 1(z)f ( i(z))]. between function spaces, then there is a weighting fuction ' and a We found that this was indeed possible to prove as we continuous bijection such that had hoped. So we now have a "nice" formula to tell us T (f ) = ' · (f ◦ ) the forms of such isometries. for all f in the function space. References Note: This function is in fact a surjective isometry and is real-linear, but it does not follow the form set by the S. Banach, \Theorie des operations linearse," Chelsea, Banach-Stone Theorem for complex-linear isometries. Warsaw, 1932. Also note that the isometry can be written in the following way: T. Miura \Surjective isometries between function T (az + b) = Re(az + b)- i Im(a(-z) + b), spaces," Contemp. Math., Vol. 645 (2015), pp. 231{240. which matches one of the conjectured forms. M. Stone, \Applications of the theory of boolean rings in topology," Trans. Amer. Math. Soc., Vol. 41 (1937), pp. 375{481. Project Advisor: Dr. Kristopher Lee 1Department of Mathematics, Iowa State University, Ames, IA 50011 ([email protected]).
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