# Functional Analysis

solution Let **Failed to parse (unknown error): **

and be metric spaces, and be a mapping.

(i) Prove that if is open whenever **Failed to parse (unknown error): **

is open, then is continuous.

(ii) Prove that **Failed to parse (unknown error): **

is continuous if and only if is closed wheneverFailed to parse (unknown error):is closed.

solution (i) Let **Failed to parse (unknown error): **

with maximum metric . Prove thatFailed to parse (unknown error):is seperable.

(ii) Use (i) to prove that **Failed to parse (unknown error): **

is also seperable with respect to the metric .

solution Assume **Failed to parse (unknown error): **

asFailed to parse (unknown error):

. Prove that **Failed to parse (unknown error): **

is continuous when is continuous for allFailed to parse (unknown error):

.

solution Show that any Cauchy sequence in a metric space is always bounded.

solution Let for **Failed to parse (unknown error): **
.

(i) Show that **Failed to parse (unknown error): **
.

(ii) Use (i) to prove that **Failed to parse (unknown error): **

(iii) Use (ii) to get **Failed to parse (unknown error): **

solution Use the inequality **Failed to parse (unknown error): **

forFailed to parse (unknown error):to prove that

**Failed to parse (unknown error): **

for **Failed to parse (unknown error): **

andFailed to parse (unknown error):

, and then use this to prove the triangular inequality:

**Failed to parse (unknown error): **

solution Assume that **Failed to parse (unknown error): **

andFailed to parse (unknown error):are two equivalent norms on , andFailed to parse (unknown error):

. Prove that is compact in **Failed to parse (unknown error): **

if and only ifFailed to parse (unknown error):is compact inFailed to parse (unknown error):

.

solution Prove the Maximum and Minimum Theorem for continuous functions on compact sets.

solution Let **Failed to parse (unknown error): **

be a closed subspace of a normed spaceFailed to parse (unknown error):

. Define **Failed to parse (unknown error): **

on the quotient spaceFailed to parse (unknown error):by

**Failed to parse (unknown error):**

for every **Failed to parse (unknown error): **
. Prove that **Failed to parse (unknown error): **

is a norm.

solution Assume that **Failed to parse (unknown error): **

is a countable linearly independent subset of a vector spaceFailed to parse (unknown error):

, and span **Failed to parse (unknown error): **
. Prove that **Failed to parse (unknown error): **

admits two inequivalent norms.

solution Let be defined by

(i) Show that is a bounded linear operator.

(ii) Find **Failed to parse (unknown error): **
.

(iii) Show that **Failed to parse (unknown error): **

is one-to-one but not onto. Find the range space ofFailed to parse (unknown error):

.

solution Prove that the dual space of **Failed to parse (unknown error): **

isFailed to parse (unknown error):

.

solution Let **Failed to parse (unknown error): **

such thatFailed to parse (unknown error):

. Show that **Failed to parse (unknown error): **

for everyFailed to parse (unknown error):

.

solution Let **Failed to parse (unknown error): **

andFailed to parse (unknown error):where 1 is in theFailed to parse (unknown error):

th position. Define **Failed to parse (unknown error): **

forFailed to parse (unknown error):

. Show that for each **Failed to parse (unknown error): **
, but **Failed to parse (unknown error): **
.

solution Let **Failed to parse (unknown error): **

be a bounded linear operator. Show thatFailed to parse (unknown error):exists and is bounded if and only if there exists such thatFailed to parse (unknown error):for every .

solution Let be a normed space and **Failed to parse (unknown error): **
. Prove that if for every bounded linear functional **Failed to parse (unknown error): **

onFailed to parse (unknown error):

, then **Failed to parse (unknown error): **
.

solution Let **Failed to parse (unknown error): **

be a closed subspace of a normed spaceFailed to parse (unknown error):

. Let **Failed to parse (unknown error): **

but not in . Use the Hahn-Banach Theorem to prove that there exists a bounded linear functionalFailed to parse (unknown error):on such that

(a) **Failed to parse (unknown error): **

for every

(b) **Failed to parse (unknown error): **

(c) **Failed to parse (unknown error): **

solution Let **Failed to parse (unknown error): **

be a bounded linear operator from a normed spaceFailed to parse (unknown error):to a normed space , and its norm be defined by

**Failed to parse (unknown error):**

Show that **Failed to parse (unknown error): **
.

solution Let **Failed to parse (unknown error): **

andFailed to parse (unknown error):be two normed spaces andFailed to parse (unknown error):be a linear mapping fromFailed to parse (unknown error):toFailed to parse (unknown error):

. Show that if **Failed to parse (unknown error): **

is continuous, then the null spaceFailed to parse (unknown error):is a closed subspace ofFailed to parse (unknown error):

. Give an example showing that the closedness of **Failed to parse (unknown error): **

does not imply the continuity ofFailed to parse (unknown error):

.

solution State the Hahn-Banach Extension theorem for bounded linear functionals on normed spaces.

solution Let **Failed to parse (unknown error): **

be a normed space. Use the Hahn-Banach Extension to prove that for everyFailed to parse (unknown error):

, there exists a bounded linear functional on **Failed to parse (unknown error): **

such that and .

solution State the Uniform Boundedness Theorem.

solution Let and be Banach spaces and . Show that the following are equivalent:

(a) is bounded;

(b) is bounded for each **Failed to parse (unknown error): **

(c) is bounded for all and all .

solution Let and define . Show that

defines a bounded linear operator on when is a continuous function on . Estimate the norm of .

solution Let and where is in the th position. Let be a sequence of complex numbers.

(a) Show that defines a bounded linear operator on if and only if . In this case, find the norm of .

(b) Find the necessary and sufficient condition for to be bounded invertible (i.e., the inverse exists and is bounded).

solution Let be a sequence of real numbers such that is convergent for every . Use the Uniform Boundedness theorem to prove that .

solution Let be a normed space and be a linearly independent subset of . Use the Hahn-Banach extension theorem to show that there exist bounded linear functionals on such that

where when and when . Can you think about extending this to an infinite sequence of vectors?