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Abstract. We show that there exist inequivalent representations of the dual space of C[0, 1] and of Lp[Rn] for p ∈ [1, ∞). We also show how these inequivalent representations reveal new and important results for both the operator and the geometric structure of these spaces. For example, if A is a proper closed subspace of C[0, 1], there always exists a closed subspace A⊥ (with the same definition as for L2[0, 1]) such that A∩A⊥ = {0} and A⊕A⊥ = C[0, 1]. Thus, the geometry of C[0, 1] can be viewed from a completely new perspective. At the operator level, we prove that every bounded linear operator A on C[0, 1] has a uniquely defined adjoint A∗ defined on C[0, 1], and both can be extended to bounded linear operators on L2[0, 1]. This leads to a polar decomposition and a spectral theorem for operators on the space. The same results also apply to Lp[Rn]. Another unexpected result is a proof of the Baire one approximation property (every closed densely defined linear operator on C[0, 1] is the limit of a sequence of bounded linear operators). A fundamental implication of this paper is that the use of inequivalent representations of the dual space is a powerful new tool for functional analysis.