Functional Analysis Answers

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.

Find the eigenvalues and eigenvectors of the matrices A =

"

1 2

−8 4#

and B =

"

a b

−b a#

.

Let p be defined on Vector Space X and satisfies p(x+y)≤p(x)+p(y) and for every scalar a, p(ax)=|a|p(x). Show that for any given x′∈X there exists a linear functional f′ on X such that f′(x′)=p(x′) and |f′(x)|≤p(x) for all x∈X.

Show that a partially ordered M can have at most one element a such that a <=x for all x in M and at most one element b such that x<=b for all xin M. [If such an a (or b) exists, it is called the least element (greatest element, respectively) of M.]

6. (Least element, greatest element) Show that a partially ordered M can have at most one element a such that a <=x for all x in M and at most one element b such that x<=b for all xin M. [If such an a (or b) exists, it is called the least element (greatest element, respectively) of M.]