Functional Analysis Answers

If (x_n) and (y_n) are sequences in the same normed space X, show that x_n→x and y_n→y implies x_n+y_n→x+y as well as αx_n→αx, where α is any scalar.

Let X and Y be normed spaces, T∈B(X,Y) and (x_n) a sequence in X. If x_n→x₀, show that Tx_n→Tx₀.

If x_n∈C[a,b] and x_n→x∈C[a,b]. Show that (x_n) is pointwise convergent on [a,b], that is, (x_n(t)) converges for every t∈C[a,b].

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.