a. Take Banache Space, E. Let us consider that L(E) is Banache space of all bounded linear operators T from E to E. This space is equipped with sup-norm.
Suppose F: L(E) into L(E) is given by:
i. F(X) = XTX^2
ii. F(X) = (X+T)^2
iii. F(X) = TX^2 + XTX + X^2T
where X inside L(E) and T: E into E is the fixed and bounded linear operator.
Calculate derivative of F at X inside L(E).
b. Take space L(R^2).
State the mapping F: L(R^2) into L(R^2) and F(X) = X^2
Calculate Jacobi Matrix of F at A, 2x2 matrix with entries (a, b, c, d).