Functional Analysis Answers

Let X and Y be metric spaces, X compact, and T: X →Y bijective

and continuous. Show that T is a homeomorphism.

If dim Y< ∞ in Riesz's lemma, show that one can even choose

 θ= 1.

Show that a compact metric space X is locally compact.

Show that R and C and, more generally, R^n and C^n are locally compact.

Give the examples of compact and noncompact curves in the plane R^2.

Show that R^n and C^n are not compact.

 Let X be a finite dimensional inner product space and T : X → X be

a linear operator. If T is self adjoint (that is < x, T x >=< T ∗x, x >).

Show that its spectrum is real. If T is unitary then show that its

eigenvalues have absolute value 1.