Let X be a normed space, z = X and f € X'. Show that T: X → X defined by T(x) = f(x)z, xe X is linear and compact.

Let V be the vector space of polynomials in & over R of degree <3 and define an inner product f. g = f₁¹ f(x)g(x)dxæ for all f, g € V. You may assume that this makes V into an inner product space. Define the linear map a: V→ V by a(f) (x) = f(x)+xƒ'(x) where the f' denotes differentiation. Which of the following holds with respect to this inner product? Select one: O a is self-adjoint O None of the others apply Ⓒa* = I-a, where I denotes the identity map O a is `anti-self adjoint' meaning a* = -a a is orthogonal

Let f: C→C be the function defined by ƒ(z) = izz. (c) Write f in the form f(x + iy) u{(2,y)+iz(x,y) where u, v : R² → R, and verify that the Cauchy-Riemann equations are satisfied if and only if x = y = 0.

Definition 28. Let V be a vector space over F (here F= R, C). Then the function (•₁•): V × V → F is said to be an inner product on V if the following conditions are satisfied: (x, x) is real and (x,x) ≥ 0 for all (x,x) = 0 if and only if x € 0₂. (i) (ii) (iii) (x, ay) = a(x, y) for all x, y V (iv) (x, y+z) = (x, y) + (x, z) for all (v) (x, y) = (y,x) for all x, y V. (a) (.,.): R¹ × R¹ → R is defined as xεν. (x, y) = x¹y, and for all x,y,z € V. Example: Verify that each of the following mappings define a real inner product. a EF. for all x, y ER".