Functional Analysis

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solution Let Failed to parse (unknown error):

and Y\, be metric spaces, and f:X\to Y\, be a mapping.

(i) Prove that if f^{{-1}}(G)\, is open whenever Failed to parse (unknown error):

is open, then f\, is continuous.

(ii) Prove that Failed to parse (unknown error):

is continuous if and only if f^{{-1}}(F)\, is closed whenever Failed to parse (unknown error): 
is closed.




solution (i) Let Failed to parse (unknown error):

with maximum metric d(f,g)=\max\{|f(t)-g(t)|:t\in [0,1]\}\,. Prove that Failed to parse (unknown error): 
is seperable.

(ii) Use (i) to prove that Failed to parse (unknown error):

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 Failed to parse (unknown error):

as Failed to parse (unknown error): 

. Prove that Failed to parse (unknown error):

is continuous when x_{n}(t)\, is continuous for all Failed to parse (unknown error): 

.




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




solution Let x_{j},y_{j}\in {\mathbb  {R}}\, for Failed to parse (unknown error): .


(i) Show that Failed to parse (unknown error): .


(ii) Use (i) to prove that Failed to parse (unknown error):


(iii) Use (ii) to get Failed to parse (unknown error):




solution Use the inequality Failed to parse (unknown error):

for Failed to parse (unknown error): 
to prove that

Failed to parse (unknown error):


for Failed to parse (unknown error):

and Failed to parse (unknown error): 

, and then use this to prove the triangular inequality:


Failed to parse (unknown error):




solution Assume that Failed to parse (unknown error):

and Failed to parse (unknown error): 
are two equivalent norms on X\,, and Failed to parse (unknown error): 

. Prove that M\, is compact in Failed to parse (unknown error):

if and only if Failed to parse (unknown error): 
is compact in Failed to parse (unknown error): 

.




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




solution Let Failed to parse (unknown error):

be a closed subspace of a normed space Failed to parse (unknown error): 

. Define Failed to parse (unknown error):

on the quotient space Failed to parse (unknown error): 
by
Failed to parse (unknown error):

for every Failed to parse (unknown error): . Prove that Failed to parse (unknown error):

is a norm.




solution Assume that Failed to parse (unknown error):

is a countable linearly independent subset of a vector space Failed to parse (unknown error): 

, and span Failed to parse (unknown error): . Prove that Failed to parse (unknown error):

admits two inequivalent norms.





FUNCTIONAL ANALYSIS BOOKS


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 Failed to parse (unknown error): .

(iii) Show that Failed to parse (unknown error):

is one-to-one but not onto. Find the range space of Failed to parse (unknown error): 

.




solution Prove that the dual space of Failed to parse (unknown error):

is Failed to parse (unknown error): 

.




solution Let Failed to parse (unknown error):

such that Failed to parse (unknown error): 

. Show that Failed to parse (unknown error):

for every Failed to parse (unknown error): 

.




solution Let Failed to parse (unknown error):

and Failed to parse (unknown error): 
where 1 is in the Failed to parse (unknown error): 

th position. Define Failed to parse (unknown error):

for Failed to parse (unknown error): 

. Show that T_{n}x\to 0\, for each Failed to parse (unknown error): , but Failed to parse (unknown error): .




solution Let Failed to parse (unknown error):

be a bounded linear operator. Show that Failed to parse (unknown error): 
exists and is bounded if and only if there exists K>0\, such that Failed to parse (unknown error): 
for every x\in X\,.




solution Let X\, be a normed space and Failed to parse (unknown error): . Prove that if f(x)=f(y)\, for every bounded linear functional Failed to parse (unknown error):

on Failed to parse (unknown error): 

, then Failed to parse (unknown error): .




solution Let Failed to parse (unknown error):

be a closed subspace of a normed space Failed to parse (unknown error): 

. Let Failed to parse (unknown error):

but not in Y\,. Use the Hahn-Banach Theorem to prove that there exists a bounded linear functional Failed to parse (unknown error): 
on X\, such that

(a) Failed to parse (unknown error):

for every y\in Y\,

(b) Failed to parse (unknown error):


(c) Failed to parse (unknown error):





solution Let Failed to parse (unknown error):

be a bounded linear operator from a normed space Failed to parse (unknown error): 
to a normed space Y\,, and its norm be defined by 
Failed to parse (unknown error):

Show that Failed to parse (unknown error): .




solution Let Failed to parse (unknown error):

and Failed to parse (unknown error): 
be two normed spaces and Failed to parse (unknown error): 
be a linear mapping from Failed to parse (unknown error): 
to Failed to parse (unknown error): 

. Show that if Failed to parse (unknown error):

is continuous, then the null space Failed to parse (unknown error): 
is a closed subspace of Failed to parse (unknown error): 

. Give an example showing that the closedness of Failed to parse (unknown error):

does not imply the continuity of Failed to parse (unknown error): 

.




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





FUNCTIONAL ANALYSIS BOOKS


solution Let Failed to parse (unknown error):

be a normed space. Use the Hahn-Banach Extension to prove that for every Failed to parse (unknown error): 

, there exists a bounded linear functional f\, on Failed to parse (unknown error):

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 Failed to parse (unknown error):

(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?



FUNCTIONAL ANALYSIS BOOKS



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