Exercise 3.55. Show that every wedge sum of Hausdorff spaces is Hausdorff.

Assume that V is a finite dimensional real inner product space and T = L(V) and self- adjoint. Show that T²+bT+cI is invertible if b² < 4c. Additionally, provide an example of a real inner product space V and T € L(V) and real numbers b, c with b² < 4c such that T² +bT+cI is not invertible. (Hint: The example shows that the hypothesis that T is self-adjoint is necessary, even for real vector spaces.)

Exercise 3.45. Let X be any space and Y be a discrete space. Show that the Cartesian product X X Y is equal to the disjoint union Lyey X, and the product topology is the same as the disjoint union topology.

Daly analysis