# Functional Analysis

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

(i) Prove that if $f^{{-1}}(G)\,$ is open whenever $G\subset Y\,$ is open, then $f\,$ is continuous.

(ii) Prove that $f\,$ is continuous if and only if $f^{{-1}}(F)\,$ is closed whenever $F\subset Y\,$ is closed.

solution (i) Let $X=C[0,1]\,$ with maximum metric $d(f,g)=\max\{|f(t)-g(t)|:t\in [0,1]\}\,$. Prove that $C[0,1]\,$ is seperable.

(ii) Use (i) to prove that $X=C[0,1]\,$ 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 $\sup \left\{|x_{n}(t)-x(t)|:t\in [0,1]\right\}\to 0\,$ as $n\to \infty \,$. Prove that $x(t)\,$ is continuous when $x_{n}(t)\,$ is continuous for all $n\,$.

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

solution Let $x_{j},y_{j}\in {\mathbb {R}}\,$ for $j=1,2,...,n\,$.

(i) Show that $\sum _{{j=1}}^{n}x_{j}y_{j}\leq \left(\sum _{{j=1}}^{n}x_{j}^{2}\right)^{{1/2}}\left(\sum _{{j=1}}^{n}y_{j}^{2}\right)^{{1/2}}\,$.

(ii) Use (i) to prove that $\sum _{{j=1}}^{n}|x_{j}y_{j}|\leq \left(\sum _{{j=1}}^{n}|x_{j}|^{2}\right)^{{1/2}}\left(\sum _{{j=1}}^{n}|y_{j}|^{2}\right)^{{1/2}}\,$

(iii) Use (ii) to get $\left(\sum _{{j=1}}^{n}|x_{j}+y_{j}|^{2}\right)^{{1/2}}\leq \left(\sum _{{j=1}}^{n}|x_{j}|^{2}\right)^{{1/2}}+\left(\sum _{{j=1}}^{n}|y_{j}|^{2}\right)^{{1/2}}\,$

solution Use the inequality $\alpha \beta \leq {\frac {\alpha ^{p}}{p}}+{\frac {\beta ^{q}}{q}}\,$ for $\alpha ,\beta \geq 0,{\frac {1}{p}}+{\frac {1}{q}}=1\,$ to prove that

$\int _{a}^{b}\left|f(t)g(t)\right|dt\leq \left(\int _{a}^{b}\left|f(t)\right|^{p}\right)^{{1/p}}\left(\int _{a}^{b}\left|g(t)\right|^{q}\right)^{{1/q}}\,$

for $p>1\,$ and ${\frac {1}{p}}+{\frac {1}{q}}=1\,$, and then use this to prove the triangular inequality:

$\left(\int _{a}^{b}\left|f(t)+g(t)\right|^{p}dt\right)^{{1/p}}\leq \left(\int _{a}^{b}\left|f(t)\right|^{p}\right)^{{1/p}}+\left(\int _{a}^{b}\left|g(t)\right|^{p}\right)^{{1/p}}\,$

solution Assume that $||\cdot ||_{1}\,$ and $||\cdot ||_{2}\,$ are two equivalent norms on $X\,$, and $M\subset X\,$. Prove that $M\,$ is compact in $(X,||\cdot ||_{1})\,$ if and only if $M\,$ is compact in $(X,||\cdot ||_{2})\,$.

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

solution Let $Y\,$ be a closed subspace of a normed space $(X,||\cdot ||_{2})\,$. Define $||\cdot ||_{0}\,$ on the quotient space $X/Y\,$ by

$||{\hat {x}}||_{0}={\mathrm (}inf)_{{x\in {\hat {x}}}}||x||\,$

for every ${\hat {x}}\in X/Y\,$. Prove that $||\cdot ||_{0}\,$ is a norm.

solution Assume that $\left\{x_{1},...,x_{n},...\right\}\,$ is a countable linearly independent subset of a vector space $X\,$, and span $\left\{x_{1},...,x_{n},...\right\}=X\,$. Prove that $X\,$ 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 $||T||\,$.

(iii) Show that $T\,$ is one-to-one but not onto. Find the range space of $T\,$.

solution Prove that the dual space of $l^{1}\,$ is $l^{\infty }\,$.

solution Let $T_{n},T\in B(X,Y)\,$ such that $||T_{n}-T||\to 0\,$. Show that $T_{n}x\to Tx\,$ for every $x\in H\,$.

solution Let $X=l^{2}\,$ and $e_{n}=(0,..,0,1,0,...)\,$ where 1 is in the $n\,$th position. Define $T_{n}x=\xi _{n}\,$ for $x=(\xi _{j})\in l^{2}\,$. Show that $T_{n}x\to 0\,$ for each $x\,$, but $||T_{n}-0||\not \to 0\,$.

solution Let $T:X\to Y\,$ be a bounded linear operator. Show that $T^{{-1}}\,$ exists and is bounded if and only if there exists $K>0\,$ such that $||Tx||\geq K||x||\,$ for every $x\in X\,$.

solution Let $X\,$ be a normed space and $x,y\in X\,$. Prove that if $f(x)=f(y)\,$ for every bounded linear functional $f\,$ on $X\,$, then $x=y\,$.

solution Let $Y\,$ be a closed subspace of a normed space $X\,$. Let $x_{0}\in X\,$ but not in $Y\,$. Use the Hahn-Banach Theorem to prove that there exists a bounded linear functional ${\hat {f}}\,$ on $X\,$ such that

(a) ${\hat {f}}(y)=0\,$ for every $y\in Y\,$

(b) $||{\hat {f}}||=1\,$

(c) ${\hat {f}}(x_{0})=d(x_{0},Y)\,$

solution Let $T\,$ be a bounded linear operator from a normed space $X\,$ to a normed space $Y\,$, and its norm be defined by

$||T||={\mathrm {sup}}\left\{||Tx||:x\in X,||x||\leq 1\right\}\,$

Show that $||T||={\mathrm {sup}}\left\{||Tx||:x\in X,||x||=1\right\}\,$.

solution Let $X\,$ and $Y\,$ be two normed spaces and $T\,$ be a linear mapping from $X\,$ to $Y\,$. Show that if $T\,$ is continuous, then the null space $N(T)\,$ is a closed subspace of $X\,$. Give an example showing that the closedness of $N(T)\,$ does not imply the continuity of $T\,$.

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

FUNCTIONAL ANALYSIS BOOKS

solution Let $X\,$ be a normed space. Use the Hahn-Banach Extension to prove that for every $x\in X\,$, there exists a bounded linear functional $f\,$ on $X\,$ 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 $x\in X\,$;

(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