<p>The aim of this paper is to study Lyapunov dimension and metric dimension of the global attractors for infinite dimensional nonlinear dynamical systems in Banach spaces. Firstly, local and global Lyapunov exponents of the linearized system are defined by the modulus of its spectrum, based on which Lyapunov dimension of attractors of the original nonlinear systems is defined. To investigate the relationship between the metric dimension and Lyapunov dimension, two covering lemmas of finite dimensional subspace of the Banach space are then established. Subsequently, the Kaplan-Yorke formula, which indicates that Hausdorff dimension of the global attractor is bounded by the Lyapunov dimension is proved. On a different note, the fractal dimension is proved to be bounded by a Lyapunov exponent related formula. At last, the abstract theoretical results are applied to a nonlinear delay differential equation (DDE) in a Banach space. Local and global Lyapunov exponents of the linearized DDE are derived as the spectrum radius of the linearized equation based on the proposed theory. Optimal estimation of Hausdorff and fractal dimension of the global attractor for the DDE are given by its Lyapunov dimension and a Lyapunov exponents related formula respectively. That is, we rigorously prove that the Kaplan-Yorke conjecture about Hausdorff dimension holds for delay equations in Banach spaces. The present work is a generalization of previous results in Hilbert spaces. In order to overcome difficulties caused by the lack of smooth inner product, the volume function, tensor product and exterior product of Banach spaces are used.</p>