Functional Analysis

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

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

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

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

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

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

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

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

solution Assume that $||\cdot ||_1\,$ 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 $i\ne j\,$ 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

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