Adjoint of an operator on a Hilbert space. Self-adjoint, Positive, Normal and Unitary operators and their properties.Projection on a Hilbert space. 

Invariance. Reducibility. Orthogonal projections. Derivatives of a continuous map from an open subset of Banach space to a Banach space. 

Rules of derivation. Derivative of a composite, Directional derivative. Mean value theorem and its applications. Partial derivatives and Jacobian Matrix. 

Continuously differentiable maps. Higher derivatives. Taylor’s formula. Inverse function theorem. Implicit function theorem. Step function, Regulated function.