Functional Analysis

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 $\displaystyle d(f,g) = \max\{|f(t)-g(t)| : t\isin[0,1]\}\,$ . 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 $\displaystyle \sup\left\{|x_n(t)-x(t)|:t\isin[0,1]\right\}\to 0\,$ 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 $\displaystyle x_j,y_j\isin\mathbb{R}\,$ for $\displaystyle j=1,2,...,n\,$ .

(i) Show that $\displaystyle \sum_{j=1}^n x_j y_j \le \left( \sum_{j=1}^n x_j^2 \right)^{1/2} \left( \sum_{j=1}^n y_j^2 \right)^{1/2}\,$ .

(ii) Use (i) to prove that $\displaystyle \sum_{j=1}^n|x_jy_j| \le \left( \sum_{j=1}^n|x_j|^2\right)^{1/2}\left( \sum_{j=1}^n|y_j|^2\right)^{1/2}\,$

(iii) Use (ii) to get $\displaystyle \left( \sum_{j=1}^n|x_j+y_j|^2 \right)^{1/2} \le \left( \sum_{j=1}^n|x_j|^2\right)^{1/2}+\left( \sum_{j=1}^n|y_j|^2\right)^{1/2}\,$

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

$\displaystyle \int_a^b \left| f(t)g(t) \right| dt \le \left( \int_a^b \left| f(t) \right|^p \right)^{1/p} \left( \int_a^b \left| g(t) \right|^q \right)^{1/q}\,$

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

$\displaystyle \left( \int_a^b \left| f(t)+g(t) \right|^p dt \right)^{1/p} \le \left( \int_a^b \left| f(t) \right|^p \right)^{1/p} + \left( \int_a^b \left| g(t) \right|^p \right)^{1/p}\,$

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

$\displaystyle ||\hat{x}||_0 = \mathrm(inf)_{x\isin\hat{x}}||x||\,$

for every $\displaystyle \hat{x}\isin X/Y\,$ . 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

$\displaystyle (Tx)(t)=\int_0^tx(\tau)d\tau\,$

(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 $\displaystyle T_n,T\isin B(X,Y)\,$ such that $\displaystyle ||T_n-T||\to 0\,$ . Show that $\displaystyle T_nx\to Tx\,$ for every $\displaystyle x\isin H\,$ .

solution Let $\displaystyle X=l^2\,$ and $\displaystyle e_n=(0,..,0,1,0,...)\,$ where 1 is in the $\displaystyle n\,$ th position. Define $\displaystyle T_nx=\xi_n\,$ for $\displaystyle x=(\xi_j)\isin l^2\,$ . Show that $\displaystyle T_nx\to 0\,$ 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 $\displaystyle ||Tx||\ge K ||x||\,$ for every $\displaystyle x\isin X\,$ .

solution Let $\displaystyle X\,$ be a normed space and $\displaystyle x,y\isin X\,$ . 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 $\displaystyle x_0\isin X\,$ 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 $\displaystyle y\isin Y\,$

(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

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

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

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 $\displaystyle x\isin X\,$ , 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 $\displaystyle T_n\isin B(X,Y)\,$ . Show that the following are equivalent:

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

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

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

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 $\displaystyle Te_n = a_ne_n (n=1,2,...)\,$ 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 $\displaystyle \left\{\alpha_n\right\}\isin l^1\,$ . Use the Uniform Boundedness theorem to prove that $\displaystyle \left\{\beta_n\right\} \isin l^\infty\,$ .

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 $\displaystyle i\ne j\,$ 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