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.


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


(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\,

(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\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.


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\,;

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


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