Adjoint of an operator on a Hilbert space: Self-adjoint, positive, normal and unitary operators and their properties, Projection on a Hilbert space.

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, Primitives and integrals, Differentiation under the integral sign, Riemann integral of function of real variable with values in normed linear space.