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Approximation with norms defined by derivations
Published online by Cambridge University Press: 09 April 2009
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A linear mapping D of the algebra of polynomial functions P[0, 1] into the algebra of all continuous complex-valued functions C[0,1] is called a derivation provided D(fg) = fD(g) + gD(f) for all polynomials f and g. The derivations of P[0, 1] into C[0,1] are easily seen to be all mappings of the form Dw where w is a continuous function on [0, 1] and Dw (f) = wf' (f' denotes the ordinary derivative of f). In fact, w = D(x) where x is the coordinate function. Let Dw be such a derivation, and let ∥ · ∥ denote the supremum norm on C[0,1]. Then Dw gives rise to an algebra norm ∥ · ∥w on P[0,1] denned by 
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- Copyright © Australian Mathematical Society 1975
References
Loy, R. J. (1970), ‘Maximal ideal spaces of Banach algebras of derivable elements’, J. Austral. Math. Soc. 11, 310–312.CrossRefGoogle Scholar
Rickart, C. E. (1960), General theory of Banach algebras (Van Nostrand, New York, 1960).Google Scholar
Sierpinski, W. (1952), General topology (University of Toronto Press, Toronto, 1952).CrossRefGoogle Scholar
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