# Tangent Directions

• B. T. Poljak
Part of the Lecture Notes in Economics and Mathematical Systems book series (LNE, volume 67)

## Abstract

The theorem of Lyusternik proved below is a powerful tool for the calculation of tangent directions. Before proceeding to a statement of the theorem, we recall the definition of a differentiable operator. Let E1, E2 be Banach spaces, P(x) an operator (generally nonlinear) with domain in E1 and range in E2. Then P(x) is said to be differentiable at a point x0 ∈ E1 if there exists a continuous linear operator A mapping E1 into E2, such that for all h ∈ E1,
$$P({x_0} + h) = P({x_0}) + Ah + r({x_0},\;h)$$
, where ‖r(x0, h)‖ = 0(‖h‖). The operator A is called the (Fréchet-)derivative of the operator P(x) and often denoted by A = P’(x0). It is clear that if E2 = R1 (i. e., P(x) is a functional), this definition coincides with the previous definition (Lecture 7) of the derivative of a functional. The derivative of an operator possesses the usual properties of derivatives (rules for differentiation of sums, composite functions, etc.). The derivative of a continuous linear operator coincides with the operator.