# Functional Analysis

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

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

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

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

(ii) Use (i) to prove that ${\displaystyle X=C[0,1]\,}$ is also seperable with respect to the metric ${\displaystyle \rho (f,g)=\left(\int _{0}^{1}|f(t)-g(t)|^{p}dt\right)^{1/p}\,}$.

solution Assume ${\displaystyle \sup \left\{|x_{n}(t)-x(t)|:t\in [0,1]\right\}\to 0\,}$ as ${\displaystyle n\to \infty \,}$. Prove that ${\displaystyle x(t)\,}$ is continuous when ${\displaystyle x_{n}(t)\,}$ is continuous for all ${\displaystyle n\,}$.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

FUNCTIONAL ANALYSIS BOOKS

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

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

(i) Show that ${\displaystyle T\,}$ is a bounded linear operator.

(ii) Find ${\displaystyle ||T||\,}$.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

FUNCTIONAL ANALYSIS BOOKS

solution Let ${\displaystyle X\,}$ be a normed space. Use the Hahn-Banach Extension to prove that for every ${\displaystyle x\in X\,}$, there exists a bounded linear functional ${\displaystyle f\,}$ on ${\displaystyle X\,}$ such that ${\displaystyle f(x)=||x||\,}$ and ${\displaystyle ||f||=1\,}$.

solution State the Uniform Boundedness Theorem.

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

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

(b) ${\displaystyle \left\{||T_{n}x||\right\}_{n=1}^{\infty }\,}$ is bounded for each ${\displaystyle x\in X\,}$;

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

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

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

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

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

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

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

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

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

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

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

FUNCTIONAL ANALYSIS BOOKS