De Gruyter Series in Nonlinear Analysis and Applications -- Title Page -- Copyright Page -- Table of Contents -- Introduction -- Acknowledgement -- Notation -- Chapter 1 - Fundamental properties of smoothness -- 1. Multilinear mappings and polynomials -- 2. Complexification -- 3. Fréchet smoothness -- 4. Taylor polynomial -- 5. Smoothness classes -- 6. Power series and their convergence -- 7. Complex mappings -- 8. Analytic mappings -- 9. Notes and remarks -- Chapter 2 - Basic properties of polynomials on ℝn -- 1. Spaces of polynomials on ℝn -- 2. Cubature formulae -- 3. Estimates related to Chebyshev polynomials -- 4. Polynomials and Lp-norms on ℝn -- 5. Polynomial identities -- 6. Estimates of coefficients of polynomials -- 7. Notes and remarks -- Chapter 3 - Weak continuity of polynomials and estimates of coefficients -- 1. Tensor products and spaces of multilinear mappings -- 2. Weak continuity and spaces of polynomials -- 3. Weak continuity and ℓ1 -- 4. (p, q)-summing operators -- 5. Estimates of coefficients of multilinear mappings -- 6. Bohr radius -- 7. Notes and remarks -- Chapter 4 - Asymptotic properties of polynomials -- 1. Finite representability and ultraproducts -- 2. Spreading models -- 3. Polynomials and p-estimates -- 4. Separating polynomials. Symmetric and sub-symmetric polynomials -- 5. Stabilisation of polynomials -- 6. Sub-symmetric polynomials on ℝn -- 7. Polynomial algebras on Banach spaces -- 8. Notes and remarks -- Chapter 5 - Smoothness and structure -- 1. Convex functions -- 2. Smooth bumps and structure I -- 3. Smooth variational principles -- 4. Smooth bumps and structure II -- 5. Local dependence on finitely many coordinates -- 6. Isomorphically polyhedral spaces -- 7. Lp spaces -- 8. C(K) spaces -- 9. Orlicz spaces -- 10. Notes and remarks -- Chapter 6 - Structural behaviour of smooth mappings.

1. Weak uniform continuity and higher smoothness -- 2. Bidual extensions -- 3. Class W -- 4. Uniformly smooth mappings from C(K), K scattered -- 5. Uniformly smooth mappings from -spaces -- 6. Fixing the canonical basis of c0 -- 7. Ranges of smooth mappings -- 8. Harmonic behaviour of smooth mappings -- 9. Notes and remarks -- Chapter 7 - Smooth approximation -- 1. Separation -- 2. Approximation by polynomials -- 3. Approximation by real-analytic mappings -- 4. Infimal convolution -- 5. Approximation of continuous mappings and partitions of unity -- 6. Non-linear embeddings into c0(Γ) -- 7. Approximation of Lipschitz mappings -- 8. Approximation of C1-smooth mappings -- 9. Approximation of norms -- 10. Notes and remarks -- Bibliography -- Notation -- De Gruyter Series in Nonlinear Analysis and Applications.