Duality in spaces of polynomials of degree at most n

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We study spaces of continuous polynomials of degree at most n between Banach spaces. Using symmetric tensor products we show that any polynomial of degree at most n has a natural linearisation and that the space of all scalar-valued polynomials of degree at most n has an isometric predual. We introduce the spaces of integral and nuclear polynomials of degree at most n and endow them with norms that allows us to develop a duality theory for spaces of polynomials of degree at most n.