By Richard A. Silverman

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The map T is a homeomorphism of Lq (0, 1) onto itself for every q ∈ (1, ∞) if p0 < p < ∞, where p0 is defined by the equation π p0 = 2π 2 . 1. 05. 2. Let p ∈ (p0 , ∞) and q ∈ (1, ∞). Then the family ( fn,p )n∈N forms a Schauder basis of Lq (0, 1) and a Riesz basis of L2 (0, 1). Proof. Since the en form a basis of Lq (0, 1) and T is a linear homeomorphism of Lq (0, 1) onto itself with Ten = f p,n (n ∈ N), it follows from [73], p. 1, p. 20 that the f n,p form a Schauder basis of Lq (0, 1). When q = 2 the argument is similar and follows [67], Sect.

As T is compact, Twk → Tw. Thus w X ≤ lim inf wk X = 1 and Tw Y = T , from k→∞ which it is immediate that w X = 1. Now take x1 = w. 3 Representations of Compact Linear Operators 21 From now on we suppose additionally that X,Y, X ∗ and Y ∗ are strictly convex. These blanket assumptions, although not always necessary, allow us to streamline the presentation. 3, given any x ∈ X\{0}, there is a unique element of X ∗ , here written as JX (x), such that JX (x) ∗ = 1 and x, JX (x) = x X ; JY is defined in a similar way.

The theory of Chap. 1 concerning the representation of compact linear maps is used to establish the existence of a countable family of certain types of weak solutions of the Dirichlet eigenvalue problem for the p-Laplacian, with associated eigenvalues. When the underlying space domain is a bounded interval in the real line more direct methods are available: we give an account of the work of [39] which leads to the representation in terms of p-trigonometric functions of the eigenfunctions of the one-dimensional p-Laplacian under a variety of initial or boundary conditions.