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\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.


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.


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?


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