Nonlinear Functional Analysis

a₁ assume B-space Banach space bilinear bounded set bundle chart closed geodesic compact component consider contained continuous mapping coordinate Corollary critical point curve defined Definition denote derivative diffeomorphism differentiable domain dt dt eigenvalues embedding equation Euclidean space exists f(xo fiber finite dimensional fixed point follows formula grad H₁ S₁ H₁(S₁ H₂ Hence Hilbert manifold Hilbert space homeomorphism homology groups homotopy hypothesis identity implicit function theorem implies inequality integral inverse Lemma let ƒ linear M₁ monotone Morse theory neighborhood Nonlinear norm obtain open set P₁ Proof properties Proposition prove real numbers Remark Riemannian manifold Riemannian metric satisfies scalar product Schwartz sequence smooth mapping submanifold subset subspace Suppose symmetric t+(p tangent space tangent vector topological space topology U₁ v₁ vanishing vector field x₁ Y₁ zero