# Functional Analysis

solution Let Failed to parse (unknown error):

and $Y\,$ be metric spaces, and $f:X\to Y\,$ be a mapping.


(i) Prove that if $f^{{-1}}(G)\,$ is open whenever Failed to parse (unknown error):

is open, then $f\,$ is continuous.


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

is continuous if and only if $f^{{-1}}(F)\,$ is closed whenever Failed to parse (unknown error):
is closed.


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

with maximum metric $d(f,g)=\max\{|f(t)-g(t)|:t\in [0,1]\}\,$. Prove that Failed 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 $\rho (f,g)=\left(\int _{0}^{1}|f(t)-g(t)|^{p}dt\right)^{{1/p}}\,$.


solution Assume Failed to parse (unknown error):

as Failed to parse (unknown error):


. Prove that Failed to parse (unknown error):

is continuous when $x_{n}(t)\,$ is continuous for all Failed to parse (unknown error):


.

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

solution Let $x_{j},y_{j}\in {\mathbb {R}}\,$ 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):

for Failed to parse (unknown error):
to prove that


Failed to parse (unknown error):

for Failed to parse (unknown error):

and Failed 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):

and Failed to parse (unknown error):
are two equivalent norms on $X\,$, and Failed to parse (unknown error):


. Prove that $M\,$ is compact in Failed to parse (unknown error):

if and only if Failed to parse (unknown error):
is compact in Failed 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 space Failed to parse (unknown error):


. Define Failed to parse (unknown error):

on the quotient space Failed 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 space Failed to parse (unknown error):


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

admits two inequivalent norms.


FUNCTIONAL ANALYSIS BOOKS

solution Let $T:C[0,1]\to C[0,1]\,$ be defined by

$(Tx)(t)=\int _{0}^{t}x(\tau )d\tau \,$

(i) Show that $T\,$ 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 of Failed to parse (unknown error):


.

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

is Failed to parse (unknown error):


.

solution Let Failed to parse (unknown error):

such that Failed to parse (unknown error):


. Show that Failed to parse (unknown error):

for every Failed to parse (unknown error):


.

solution Let Failed to parse (unknown error):

and Failed to parse (unknown error):
where 1 is in the Failed to parse (unknown error):


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

for Failed to parse (unknown error):


. Show that $T_{n}x\to 0\,$ 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 that Failed to parse (unknown error):
exists and is bounded if and only if there exists $K>0\,$ such that Failed to parse (unknown error):
for every $x\in X\,$.


solution Let $X\,$ be a normed space and Failed to parse (unknown error): . Prove that if $f(x)=f(y)\,$ for every bounded linear functional Failed to parse (unknown error):

on Failed to parse (unknown error):


, then Failed to parse (unknown error): .

solution Let Failed to parse (unknown error):

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


. Let Failed to parse (unknown error):

but not in $Y\,$. Use the Hahn-Banach Theorem to prove that there exists a bounded linear functional Failed to parse (unknown error):
on $X\,$ such that


(a) Failed to parse (unknown error):

for every $y\in Y\,$


(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 space Failed to parse (unknown error):
to a normed space $Y\,$, 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):

and Failed to parse (unknown error):
be two normed spaces and Failed to parse (unknown error):
be a linear mapping from Failed to parse (unknown error):
to Failed to parse (unknown error):


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

is continuous, then the null space Failed to parse (unknown error):
is a closed subspace of Failed to parse (unknown error):


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

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


.

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

FUNCTIONAL ANALYSIS BOOKS

solution Let Failed to parse (unknown error):

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


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

such that $f(x)=||x||\,$ and $||f||=1\,$.


solution State the Uniform Boundedness Theorem.

solution Let $X\,$ and $Y\,$ be Banach spaces and $T_{n}\in B(X,Y)\,$. Show that the following are equivalent:

(a) $\left\{||T_{n}||\right\}_{{n=1}}^{\infty }\,$ is bounded;

(b) $\left\{||T_{n}x||\right\}_{{n=1}}^{\infty }\,$ is bounded for each Failed to parse (unknown error):

(c) $\left\{g\left(T_{n}x\right)||\right\}_{{n=1}}^{\infty }\,$ is bounded for all $x\in X\,$ and all $g\in Y'\,$.

solution Let $L^{2}[a,b]=\left\{f:\int _{a}^{b}\left|f(t)\right|^{2}dt<\infty \right\}\,$ and define $||f||=\left(\int _{a}^{b}\left|f(t)\right|^{2}dt\right)^{{1/2}}\,$. Show that

$(Tf)(t)=\int _{a}^{b}K(s,t)f(s)ds\,$

defines a bounded linear operator on $L^{2}[a,b]\,$ when $K(s,t)\,$ is a continuous function on $[a,b]\times [a,b]\,$. Estimate the norm of $T\,$.

solution Let $H=l^{2}\,$ and $e_{n}=(0,...,0,1,0,...)\,$ where $1\,$ is in the $n\,$th position. Let $\{a_{n}\}\,$ be a sequence of complex numbers.

(a) Show that $Te_{n}=a_{n}e_{n}(n=1,2,...)\,$ defines a bounded linear operator on $H\,$ if and only if ${\mathrm {sup}}\left\{|a_{n}|:n=1,2,...\right\}<\infty \,$. In this case, find the norm of $T\,$.

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

solution Let $\left\{\beta _{n}\right\}\,$ be a sequence of real numbers such that $\sum _{{n=1}}^{\infty }\alpha _{n}\beta _{n}\,$ is convergent for every $\left\{\alpha _{n}\right\}\in l^{1}\,$. Use the Uniform Boundedness theorem to prove that $\left\{\beta _{n}\right\}\in l^{\infty }\,$.

solution Let $X\,$ be a normed space and $\{x_{1},...,x_{n}\}\,$ be a linearly independent subset of $X\,$. Use the Hahn-Banach extension theorem to show that there exist bounded linear functionals $f_{1},...,f_{n}\,$ on $X\,$ such that

$f_{i}(x_{j})=\delta _{{i,j}}\,$

where $\delta _{{i,j}}=0\,$ when $i\neq j\,$ and $1\,$ when $i=j\,$. Can you think about extending this to an infinite sequence $\left\{x_{1},...,x_{n},...\right\}\,$ of vectors?

FUNCTIONAL ANALYSIS BOOKS