# Tangent Directions

Chapter

## 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 E, where ‖r(x

_{1}, E_{2}be Banach spaces, P(x) an operator (generally nonlinear) with domain in E_{1}and range in E_{2}. Then P(x) is said to be differentiable at a point x_{0}∈ E_{1}if there exists a continuous linear operator A mapping E_{1}into E_{2}, such that for all h ∈ E_{1},$$P({x_0} + h) = P({x_0}) + Ah + r({x_0},\;h)$$

_{0}, h)‖ = 0(‖h‖). The operator A is called the (Fréchet-)derivative of the operator P(x) and often denoted by A = P’(x_{0}). It is clear that if E_{2}= R^{1}(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.## Preview

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## Copyright information

© Springer-Verlag Berlin · Heidelberg 1972