a) Let X={f in C[0,1]:f(0)=0}X=\{f \in C[0,1]: f(0)=0\}
Y={g in x:int_(0)^(1)g(t)dt=0}Y=\left\{g \in x: \int_0^1 g(t) d t=0\right\}
Prove that Y\mathrm{Y} is a proper subspace of X\mathrm{X}. Is Y\mathrm{Y} a closed subspace of X\mathrm{X} ? Justify your answer.
b) Let X=L^(p)[0,1]X=L^p[0,1] and x=x(t)=t^(2)x=x(t)=t^2. Find ||x||_(p)\|x\|_p for p=4p=4 and oo\infty.
c) Let E be a subset of a normed space X,Y=span EX, Y=\operatorname{span} E and a in Xa \in X. Show that a in bar(Y)a \in \bar{Y} if and only if f(a)=0f(a)=0 whenever f inX^(‘)f \in X^{\prime} and f=0f=0 everywhere on EE.
2. a) Consider the space c_(00)c_{00}. For x=(x_(1),x_(2),dots,x_(n),dots)inc_(00)x=\left(x_1, x_2, \ldots, x_n, \ldots\right) \in c_{00}, define f(x)=sum_(n=1)^(oo)x_(n)f(x)=\sum_{n=1}^{\infty} x_n. Show that ff is a linear functional which is not continuous w.r.t the norm ||x||=s u p _(n)|x_(n)|\|x\|=\sup _n\left|x_n\right|.
b) Consider the space C^(1)[0,1]\mathrm{C}^1[0,1] of all C^(1)\mathrm{C}^1 functions on [0,1] endowed with the uniform norm induced from the space C[0,1]\mathrm{C}[0,1], and consider the differential operator D:(C^(1)[0,1],||*||_(oo))rarr(C[0,1],||*||_(oo))D:\left(C^1[0,1],\|\cdot\|_{\infty}\right) \rightarrow\left(C[0,1],\|\cdot\|_{\infty}\right) defined by Df=f^(‘)D f=f^{\prime}. Prove that DD is linear, with closed graph, but not continuous. Can we conclude from here that C^(1)[0,1]C^1[0,1] is not a Banach space? Justify your answer.
3. a) When is a normed linear space called separable? Show that a normed linear space is separable if its dual is separable [You should state all the proposition or theorems or corollaries used for proving the theorem]. Is the converse true? Give justification for your answer. [Whenever an example is given, you should justify that the example satisfies the requirements.]
b) Let X\mathrm{X} be a Banach space, Y\mathrm{Y} be a normed linear space and F\mathcal{F} be a subset of B(X,Y)\mathrm{B}(\mathrm{X}, \mathrm{Y}). If F\mathcal{F} is not uniformly bounded, then there exists a dense subset D\mathrm{D} of X\mathrm{X} such that for every xinD,{F(x):FinF}\mathrm{x} \in \mathrm{D},\{\mathrm{F}(\mathrm{x}): \mathrm{F} \in \mathcal{F}\} is not bounded in Y\mathrm{Y}.
4. a) Read the proof of the closed graph theorem carefully and explain where and how we have used the following facts in the proof.
i) X\mathrm{X} is a Banach space.
ii) Y\mathrm{Y} is a Banach space.
iii) F is a closed map.
iv) Which property of continuity is being established to conclude that F\mathrm{F} is continuous.
b) Which of the following maps are open? Give reasons for your answer.
i) quadT:R^(3)rarrR^(2)\quad \mathrm{T}: \mathbb{R}^3 \rightarrow \mathbb{R}^2 given by T(x,y,z)=(x,z)\mathrm{T}(\mathrm{x}, \mathrm{y}, \mathrm{z})=(\mathrm{x}, \mathrm{z}).
ii) quadT:R^(3)rarrR^(3)\quad \mathrm{T}: \mathbb{R}^3 \rightarrow \mathbb{R}^3 given by T(x,y,z)=(x,y,0)\mathrm{T}(\mathrm{x}, \mathrm{y}, \mathrm{z})=(\mathrm{x}, \mathrm{y}, 0).
5. a) Let f:C[0,1]rarrRf: C[0,1] \rightarrow \mathbb{R} be given by f(x)=x(1)AA x in C[0,1]f(x)=x(1) \forall x \in C[0,1]. Show that ff is continuous w.r.t the supnorm and ff is not continuous w.r.t the p-norm.
b) Let X\mathrm{X} be an inner product space and x,yinX\mathrm{x}, \mathrm{y} \in \mathrm{X}. Prove that x_|_y\mathrm{x} \perp \mathrm{y} if and only if ||kx+y||^(2)=||kx||^(2)+||y^(2)||,kinK\|\mathrm{kx}+\mathrm{y}\|^2=\|\mathrm{kx}\|^2+\left\|\mathrm{y}^2\right\|, \mathrm{k} \in \mathrm{K}.
6. a) Let H=R^(3)\mathrm{H}=\mathrm{R}^3 and F\mathrm{F} be the set of all x=(x_(1),x_(2),x_(3))\mathbf{x}=\left(\mathrm{x}_1, \mathrm{x}_2, \mathrm{x}_3\right) in H\mathrm{H} such that x_(1)=0\mathrm{x}_1=0. Find F^(_|_)\mathrm{F}^{\perp}. Verify that every xinH\mathbf{x} \in \mathrm{H} can be expressed as x=y+z\mathbf{x}=\mathbf{y}+\mathbf{z} where yinF\mathbf{y} \in \mathrm{F} and zinF^(_|_)\mathbf{z} \in \mathrm{F}^{\perp}.
b) Given an example of an Hilbert space H\mathrm{H} and an operator A\mathrm{A} on H\mathrm{H} such that sigma_(e)(A)\sigma_{\mathrm{e}}(\mathrm{A}) is empty. Justify your choice of example.
c) Let AA be a normal operator on a Hilbert space XX. Show that sigma(A)subsigma _(a)(A)\sigma(A) \subset \sigma_a(A) where sigma _(a)(A)\sigma_a(A) denotes the approximate eigen spectrum of AA and sigma(A)\sigma(A) denotes the spectrum of A.
7. a) Let X=c_(00)X=c_{00} with ||*||_(p)\|\cdot\|_p. Give an example of a Cauchy sequence in XX that do not converge in X. Justify your choice of example.
b) Give one example of each of the following. Also justify your choice of example.
i) A self-adjoint operator on ℓ^(2)\ell^2.
ii) A normal operator on a Hilbert space which is not unitary.
c) Let X\mathrm{X} be a normed space and Y\mathrm{Y} be proper subspace of X\mathrm{X}. Show that the interior Y^(0)\mathrm{Y}^0 of Y\mathrm{Y} is empty.
8. a) Let X,YX, Y be normed spaces and suppose BL(X,Y) and CL(X,Y) denote, respectively, the space of bounded linear operators from X\mathrm{X} to Y\mathrm{Y} and the space of compact linear operators from X to Y. Show that CL(X,Y) is linear subspace of BL(X,Y). Also, Show that if Y\mathrm{Y} is a Banach space, then CL(X,Y)\mathrm{CL}(\mathrm{X}, \mathrm{Y}) is a closed subspace of BL(X,Y)\mathrm{BL}(\mathrm{X}, \mathrm{Y}).
b) Define a Hilbert-Schmidt operator on a Hilbert space H\mathrm{H} and give an example. Is every Hilbert-sehmidt operator a compact operator? Justify your answer.
9. a) Let {A_(n)}\left\{A_n\right\} be a sequence of unitary operators in BL(H)B L(H). Prove that if ||A_(n)-A||rarr0,A in BL(H)\left\|A_n-A\right\| \rightarrow 0, A \in B L(H), then AA is unitary.
b) Define the spectral radius of a bounded linear operator A in BL(X)A \in B L(X). Find the spectral radius of A\mathrm{A} in BL(R^(3))\mathrm{BL}\left(\mathbb{R}^3\right), where A\mathrm{A} is given by the matrix
with respect to the standard basis of R^(3)\mathbb{R}^3.
c) Let X\mathrm{X} be a Banach space and Y\mathrm{Y} be a closed subspace of X\mathrm{X}. Let pi:XrarrX//Y\pi: \mathrm{X} \rightarrow \mathrm{X} / \mathrm{Y} be canonical quotient map. Show that pi\pi is open.
10. State giving reasons, if the following statement are true or false.
a) A closed map on a normed space need not be an open map.
b) c_(00)\mathrm{c}_{00} is a closed subspace of ℓ^(oo)\ell^{\infty}.
c) The dual of a finite dimensional space is finite dimensional.
d) If T_(1)T_1 and T_(2)T_2 are positive operators on a Hilbert space HH, then T_(1)+T_(2)T_1+T_2 is a positive operator on H\mathrm{H}.
e) On a normed space X\mathrm{X}, the norm function ||*||:XrarrC\|\cdot\|: \mathrm{X} \rightarrow \mathbb{C} is a linear map.
a) Let X={f in C[0,1]:f(0)=0}X=\{f \in C[0,1]: f(0)=0\}
Y={g in x:int_(0)^(1)g(t)dt=0}Y=\left\{g \in x: \int_0^1 g(t) d t=0\right\}
Prove that Y\mathrm{Y} is a proper subspace of X\mathrm{X}. Is Y\mathrm{Y} a closed subspace of X\mathrm{X} ? Justify your answer.
Answer:
To prove that YY is a proper subspace of XX and determine whether YY is a closed subspace of XX, we need to examine the definitions and properties of these sets within the context of functional analysis.
XX and YY Defined
X={f in C[0,1]:f(0)=0}X = \{f \in C[0,1]: f(0) = 0\} is the set of all continuous functions on the interval [0,1][0,1] that vanish at 00.
Y={g in X:int_(0)^(1)g(t)dt=0}Y = \left\{g \in X: \int_0^1 g(t) dt = 0\right\} is the set of all functions in XX whose integral over [0,1][0,1] is 00.
Proving YY is a Proper Subspace of XX
To show that YY is a proper subspace of XX, we must verify that YY satisfies the following criteria for being a subspace:
Non-emptiness: YY contains the zero function, which is the function g(t)=0g(t) = 0 for all t in[0,1]t \in [0,1]. This function clearly belongs to XX and satisfies the integral condition, so YY is non-empty.
Closed under addition: If g_(1),g_(2)in Yg_1, g_2 \in Y, then g_(1)+g_(2)in Yg_1 + g_2 \in Y. This is because the integral of the sum is the sum of the integrals, each of which is 00, so their sum is also 00.
Closed under scalar multiplication: If g in Yg \in Y and alpha\alpha is a scalar, then alpha g in Y\alpha g \in Y. This follows because int_(0)^(1)alpha g(t)dt=alphaint_(0)^(1)g(t)dt=alpha*0=0\int_0^1 \alpha g(t) dt = \alpha \int_0^1 g(t) dt = \alpha \cdot 0 = 0.
Since YY satisfies these criteria, it is a subspace of XX. It is a proper subspace because there exist functions in XX that do not satisfy the integral condition, such as f(t)=tf(t) = t, which is in XX but not in YY since int_(0)^(1)tdt=1//2!=0\int_0^1 t dt = 1/2 \neq 0.
Is YY a Closed Subspace of XX?
A subspace YY is closed in XX if it contains all its limit points; that is, if a sequence of functions {g_(n)}\{g_n\} in YY converges uniformly to a function gg, then gg must also be in YY.
To show YY is closed, consider a sequence {g_(n)}sub Y\{g_n\} \subset Y that converges uniformly to g in Xg \in X. We need to show that g in Yg \in Y, meaning int_(0)^(1)g(t)dt=0\int_0^1 g(t) dt = 0.
Uniform convergence of {g_(n)}\{g_n\} to gg implies that for every epsilon > 0\epsilon > 0, there exists an NN such that for all n >= Nn \geq N, we have |g_(n)(t)-g(t)| < epsilon|g_n(t) – g(t)| < \epsilon for all t in[0,1]t \in [0,1]. By the properties of integrals and the limit of a sequence of functions,
Since each g_(n)in Yg_n \in Y, we have int_(0)^(1)g_(n)(t)dt=0\int_0^1 g_n(t) dt = 0. Thus, the limit of these integrals as n rarr oon \to \infty is also 00, which means int_(0)^(1)g(t)dt=0\int_0^1 g(t) dt = 0, and hence g in Yg \in Y.
Therefore, YY is a closed subspace of XX because it contains all its limit points under uniform convergence.
b) Let X=L^(p)[0,1]X=L^p[0,1] and x=x(t)=t^(2)x=x(t)=t^2. Find ||x||_(p)\|x\|_p for p=4p=4 and oo\infty.
Answer:
To find the L^(p)L^p norm of x(t)=t^(2)x(t) = t^2 on the interval [0,1][0,1] for p=4p=4 and p=oop=\infty, we’ll use the definitions of the L^(p)L^p norms.
To compute this integral, we use the formula for the integral of t^(n)t^n, which is (t^(n+1))/(n+1)\frac{t^{n+1}}{n+1} for n!=-1n \neq -1, from 00 to 11:
The L^( oo)L^\infty norm, or the supremum norm, is defined as the essential supremum of |x(t)||x(t)| over the interval. For x(t)=t^(2)x(t) = t^2 on [0,1][0,1], this is simply the maximum value of t^(2)t^2 on the interval, which occurs at t=1t=1. Therefore,
For p=4p=4, ||x||_(4)=(1)/(sqrt3)\|x\|_4 = \frac{1}{\sqrt{3}}.
For p=oop=\infty, ||x||_(oo)=1\|x\|_\infty = 1.
c) Let E be a subset of a normed space X,Y=span EX, Y=\operatorname{span} E and a in Xa \in X. Show that a in bar(Y)a \in \bar{Y} if and only if f(a)=0f(a)=0 whenever f inX^(‘)f \in X^{\prime} and f=0f=0 everywhere on EE.
Answer:
To show that a in bar(Y)a \in \bar{Y} if and only if f(a)=0f(a) = 0 for every f inX^(‘)f \in X’ (the dual space of XX) where f=0f = 0 on EE, we’ll use some fundamental properties of normed spaces, their duals, and the concept of closure.
Definitions:
XX is a normed space, and X^(‘)X’ is its dual space, consisting of all continuous linear functionals on XX.
EE is a subset of XX, and Y=span EY = \operatorname{span}E is the subspace spanned by EE.
bar(Y)\bar{Y} denotes the closure of YY in XX, which includes all limits of convergent sequences in YY.
(=>\Rightarrow) If a in bar(Y)a \in \bar{Y}, then f(a)=0f(a) = 0 for every f inX^(‘)f \in X’ where f=0f = 0 on EE:
Assume a in bar(Y)a \in \bar{Y}. This means that for every epsilon > 0\epsilon > 0, there exists y in Yy \in Y such that ||a-y|| < epsilon\|a – y\| < \epsilon, since aa can be approximated arbitrarily closely by elements of YY.
Let f inX^(‘)f \in X’ be such that f=0f = 0 on EE. Since Y=span EY = \operatorname{span}E, ff also vanishes on YY because any linear combination of elements in EE (which ff maps to 00) will also be mapped to 00 by ff.
Given a in bar(Y)a \in \bar{Y}, and considering ff is continuous, we have f(a)=lim_(y rarr a,y in Y)f(y)f(a) = \lim_{y \to a, y \in Y} f(y). Since f(y)=0f(y) = 0 for all y in Yy \in Y, it follows that f(a)=0f(a) = 0.
(lArr\Leftarrow) If f(a)=0f(a) = 0 for every f inX^(‘)f \in X’ where f=0f = 0 on EE, then a in bar(Y)a \in \bar{Y}:
Assume f(a)=0f(a) = 0 for every f inX^(‘)f \in X’ where f=0f = 0 on EE. Suppose, for the sake of contradiction, that a!in bar(Y)a \notin \bar{Y}. By the Hahn-Banach theorem, there exists a continuous linear functional f inX^(‘)f \in X’ such that f(a)!=0f(a) \neq 0 and f=0f = 0 on bar(Y)\bar{Y} (since aa and bar(Y)\bar{Y} can be separated).
However, since f=0f = 0 on bar(Y)\bar{Y}, and E sube Y sube bar(Y)E \subseteq Y \subseteq \bar{Y}, it follows that f=0f = 0 on EE. But we assumed that f(a)=0f(a) = 0 for all such ff, contradicting the existence of such an ff that separates aa from bar(Y)\bar{Y}.
Therefore, our assumption that a!in bar(Y)a \notin \bar{Y} must be false, implying a in bar(Y)a \in \bar{Y}.
Conclusion:
We’ve shown that a in bar(Y)a \in \bar{Y} if and only if f(a)=0f(a) = 0 for every f inX^(‘)f \in X’ where f=0f = 0 on EE, using the properties of normed spaces, their duals, and the Hahn-Banach theorem for the separation argument.
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