a) Let X=C[0,1]\mathrm{X}=\mathrm{C}[0,1]. Define d:X xx X rarrRd: X \times X \rightarrow \mathbf{R} by d(f,g)=int_(0)^(1)|f(t)-g(t)|dt,f,ginX\mathrm{d}(\mathrm{f}, \mathrm{g})=\int_0^1|\mathrm{f}(\mathrm{t})-\mathrm{g}(\mathrm{t})| \mathrm{dt}, \mathrm{f}, \mathrm{g} \in \mathrm{X} where the integral is the Riemann integral. Show that dd is a metric on XX. Find d(f,g)d(f, g) where f(x)=4xf(x)=4 x and g(x)=x^(3),x in[0,1]g(x)=x^3, x \in[0,1].
b) Let ( X,d)X, d) be a metric space and a in Xa \in X be a fixed point of XX. Show that the function f_(a):X rarrRf_a: X \rightarrow \mathbf{R} given by f_(a)(x)=d(x,a)\mathrm{f}_{\mathrm{a}}(\mathrm{x})=\mathrm{d}(\mathrm{x}, \mathrm{a}) is continuous. Is it uniformly continuous? Justify you answer.
2. a) Let AA and BB be any two subsets of a metric space (X, d), then show that
i) int A=uu{E\mathrm{A}=\cup\{\mathrm{E} : is open and EsubeA}\mathrm{E} \subseteq \mathrm{A}\}
ii) int(AnnB)=int Ann int B\operatorname{int}(\mathrm{A} \cap \mathrm{B})=\operatorname{int} \mathrm{A} \cap \operatorname{int} \mathrm{B}
iii) quad int(A uu B)supe\quad \operatorname{int}(A \cup B) \supseteq int A nnA \cap int BB
iv) bar(AnnB)sube bar(A)nn bar(B)\overline{\mathrm{A} \cap \mathrm{B}} \subseteq \overline{\mathrm{A}} \cap \overline{\mathrm{B}}.
b) Find the interior, boundary and closure of the following sets A\mathbf{A} in R\mathbf{R} with the usual metric and discrete metric.
i) A=QA=\mathbf{Q}, the set of rationals in R\mathbf{R}
ii) A=]1,2]uu]2,4[\mathrm{A}=] 1,2] \cup] 2,4[
3. a) Let (X,d_(1))\left(X, d_1\right) and (Y,d_(2))\left(Y, d_2\right) be metric spaces. Show that f:X rarr Yf: X \rightarrow Y is continuous if and only if f( bar(A))sube bar(f(A))f(\bar{A}) \subseteq \overline{f(A)} where AA is any subset of XX
b) Let (X_(1),d_(1))\left(\mathrm{X}_1, \mathrm{~d}_1\right) and (X_(2),d_(2))\left(\mathrm{X}_2, \mathrm{~d}_2\right) be two discrete metric spaces. Verify that the product metric on X_(1)xxX_(2)\mathrm{X}_1 \times \mathrm{X}_2 is discrete.
c) Show that an infinite discrete metric space X\mathrm{X} is bounded but not totally bounded.
4. a) Find the first derivative f^(‘)(a)\mathrm{f}^{\prime}(\mathbf{a}) of the function f\mathrm{f} defined by f:R^(3)rarrR^(2)f: \mathbf{R}^3 \rightarrow \mathbf{R}^2 given by f(x,y,z)=(xyz,x+y+z^(2))f(x, y, z)=\left(x y z, x+y+z^2\right) where a=(1.-1,2)\mathbf{a}=(1 .-1,2).
b) Let E\mathrm{E} be an open subset of R^(n)\mathbf{R}^n and f:E rarrR^(m)f: E \rightarrow \mathbf{R}^m be a function such that each of its components function f_(i)f_i are differentiable, then show that ff is differentiable. Is the converse of this result true? Justify your answer.
c) Near what points may the surface z^(2)+xz+y=0z^2+x z+y=0 be represented uniquely as a graph of a differentiable function z=k(x,y)\mathrm{z}=\mathrm{k}(\mathrm{x}, \mathrm{y}) ? Locate such a point.
5. a) Use the method of Lagrange’s multiplier method to find the shortest possible distance from the ellipse x^(2)+2y^(2)=2x^2+2 y^2=2 to the line x+y=2x+y=2.
b) Find the directional derivative of the function f:R^(4)rarrR^(3)f: \mathbf{R}^4 \rightarrow \mathbf{R}^3 defined by
f(x,y,z,w)=(x^(2)y,xyz,x^(2)+y^(2)+z^(2))f(x, y, z, w)=\left(x^2 y, x y z, x^2+y^2+z^2\right)
at a=(1,2,-1,-2)\mathrm{a}=(1,2,-1,-2) in the direction v=(0,1,2,-2)\mathrm{v}=(0,1,2,-2).
6. a) Let A be a compact non-empty subset of a metric space (X, d) and let F be a closed subset of XX such that A nn F=phiA \cap F=\phi, then show that d(A,F) > 0d(A, F)>0 where d(A,F)=i n f{d(a,b):a in A,b in F}d(A, F)=\inf \{d(a, b): a \in A, b \in F\}.
b) Give an example of the following with justification
i) A vector-valued function f:R^(3)rarrR^(3)f: \mathbf{R}^3 \rightarrow \mathbf{R}^3 which is not differentiable at (0,0,0)(0,0,0).
ii) A function which is Legesgue measurable on R\mathbf{R}.
c) Show that the components of a metric space is either identical or pairwise disjoint.
7. a) Let Q\mathbf{Q} be the set of rationals with the metric defined on Q\mathbf{Q} by d:QxxQrarrRd: \mathbf{Q} \times \mathbf{Q} \rightarrow \mathbf{R}, defined by d(x,y)=|x-y|,AA x,y inRd(x, y)=|x-y|, \forall x, y \in \mathbf{R}.
Show that {(1+(1)/(n))^(n)}\left\{\left(1+\frac{1}{\mathrm{n}}\right)^{\mathrm{n}}\right\} is Cauchy sequence in Q\mathbf{Q}, but does not converge in Q\mathbf{Q} and {(1)/(3^(n))}\left\{\frac{1}{3^n}\right\} is a Cauchy sequence Q\mathbf{Q} which converges in Q\mathbf{Q} to the limit 0 .
b) Which of the following sets are totally bounded? Give reasons for your answer. Are they compact?
i) quad2N\quad 2 \mathbf{N} in (N,d)(\mathbf{N}, d) where dd is the discrete metric.
ii) quad[0,2]uu[5,10]\quad[0,2] \cup[5,10] in (R,d)(\mathbf{R}, d) where dd is the Euclidean metric.
c) Which of the following sets are connected sets in R^(2)\mathbf{R}^2 with the metric given against it? Justify your answer.
i) quadA={(x,y):0 <= x <= 1,0 <= y <= 2}\quad \mathrm{A}=\{(\mathrm{x}, \mathrm{y}): 0 \leq \mathrm{x} \leq 1,0 \leq \mathrm{y} \leq 2\} under the standard metric.
ii) A={(x,y):x^(2)+y^(2)=1}\mathrm{A}=\left\{(\mathrm{x}, \mathrm{y}): \mathrm{x}^2+\mathrm{y}^2=1\right\} under the discrete metric.
8. a) Consider Z\mathbf{Z} and let F_(1)\mathcal{F}_1 denote the class of subsets of Z\mathbf{Z}, given by F_(1)={AsubZ\mathcal{F}_1=\{\mathrm{A} \subset \mathbf{Z} : either A\mathrm{A} is finite or A^(c)\mathrm{A}^{\mathrm{c}} is finite }. Check whether F_(1)\mathcal{F}_1 is a sigma\sigma algebra or not.
b) Let A be any set in R\mathbf{R}, show that m^(**)(A)=m^(**)(A+x)m^*(A)=m^*(A+x) where m^(**)m^* denotes the outer measure.
c) Find the measure of the following sets.
i) quad E=nnn_(n=1)^(oo)(a-(1)/(n),b)\quad E=\bigcap_{n=1}^{\infty}\left(a-\frac{1}{n}, b\right)
ii) E=Quu{1,2,3,4}\mathrm{E}=\mathbf{Q} \cup\{1,2,3,4\}
iii) E=]5,7[uu[7,7.5]\mathrm{E}=] 5,7[\cup[7,7.5].
9. a) Show that if ff is measurable, then the function f^(a)(x)f^a(x) given by
f^(a)(x)={[a,” if “f(x) > a],[f(x),” if “f(x) <= a]:}f^a(x)=\left\{\begin{array}{cc}
a & \text { if } f(x)>a \\
f(x) & \text { if } f(x) \leq a
\end{array}\right.
is also measurable.
b) Verify Bounded Convergence Theorem for the sequence of functions {f_(n)}\left\{f_n\right\} where
c) Find the fourier series of the function ff defined by
f(x)={[-x^(2)”,”-pi < x <= 0],[x^(2)”,”0 < x < pi]:}f(x)=\left\{\begin{array}{c}
-x^2,-\pi<x \leq 0 \\
x^2, 0<x<\pi
\end{array}\right.
State whether the following statements are True or False. Justify your answers.
a) The sequence {((1)/(n),(1)/(n)):ninN}\left\{\left(\frac{1}{\mathrm{n}}, \frac{1}{\mathrm{n}}\right): \mathrm{n} \in \mathbf{N}\right\} is convergent in R^(2)\mathbf{R}^2 under the discrete metric on R^(2)\mathbf{R}^2.
b) A subset in a metric space is compact if it is closed.
c) Continuous image of a path connected space is path connected.
d) The second derivative of a linear map from R^(n)\mathbf{R}^n to R^(m)\mathbf{R}^m never vanishes.
e) If int_(A)fdm=int_(A)gdm\int_{\mathrm{A}} \mathrm{fdm}=\int_{\mathrm{A}} \mathrm{gdm} for all Ain M\mathrm{A} \in \boldsymbol{M}, then f=g\mathrm{f}=\mathrm{g}.
a) Let X=C[0,1]\mathrm{X}=\mathrm{C}[0,1]. Define d:X xx X rarrRd: X \times X \rightarrow \mathbf{R} by d(f,g)=int_(0)^(1)|f(t)-g(t)|dt,f,ginX\mathrm{d}(\mathrm{f}, \mathrm{g})=\int_0^1|\mathrm{f}(\mathrm{t})-\mathrm{g}(\mathrm{t})| \mathrm{dt}, \mathrm{f}, \mathrm{g} \in \mathrm{X} where the integral is the Riemann integral. Show that dd is a metric on XX. Find d(f,g)d(f, g) where f(x)=4xf(x)=4 x and g(x)=x^(3),x in[0,1]g(x)=x^3, x \in[0,1].
Answer:
To show that dd is a metric on XX, we need to verify the following properties for all f,g,h in Xf, g, h \in X:
Non-negativity:d(f,g) >= 0d(f, g) \geq 0
Identity of indiscernibles:d(f,g)=0d(f, g) = 0 if and only if f=gf = g
Non-negativity:
For any f,g in Xf, g \in X, the absolute value function |*||\cdot| ensures that |f(t)-g(t)| >= 0|f(t) – g(t)| \geq 0 for all t in[0,1]t \in [0, 1]. Therefore, the integral int_(0)^(1)|f(t)-g(t)|dt\int_0^1 |f(t) – g(t)| \, dt is also non-negative. Hence, d(f,g) >= 0d(f, g) \geq 0.
Identity of indiscernibles:
If f=gf = g, then f(t)-g(t)=0f(t) – g(t) = 0 for all t in[0,1]t \in [0, 1], so d(f,g)=int_(0)^(1)|f(t)-g(t)|dt=int_(0)^(1)0dt=0d(f, g) = \int_0^1 |f(t) – g(t)| \, dt = \int_0^1 0 \, dt = 0.
Conversely, if d(f,g)=0d(f, g) = 0, then int_(0)^(1)|f(t)-g(t)|dt=0\int_0^1 |f(t) – g(t)| \, dt = 0. Since the integrand is non-negative, it must be zero almost everywhere, implying that f(t)=g(t)f(t) = g(t) for almost all t in[0,1]t \in [0, 1]. Since ff and gg are continuous, they must be equal everywhere on [0,1][0, 1], so f=gf = g.
Symmetry:
By the properties of the absolute value function, |f(t)-g(t)|=|g(t)-f(t)||f(t) – g(t)| = |g(t) – f(t)| for all t in[0,1]t \in [0, 1]. Therefore, d(f,g)=int_(0)^(1)|f(t)-g(t)|dt=int_(0)^(1)|g(t)-f(t)|dt=d(g,f)d(f, g) = \int_0^1 |f(t) – g(t)| \, dt = \int_0^1 |g(t) – f(t)| \, dt = d(g, f).
Triangle inequality:
For any f,g,h in Xf, g, h \in X and for all t in[0,1]t \in [0, 1], we have |f(t)-h(t)|=|f(t)-g(t)+g(t)-h(t)| <= |f(t)-g(t)|+|g(t)-h(t)||f(t) – h(t)| = |f(t) – g(t) + g(t) – h(t)| \leq |f(t) – g(t)| + |g(t) – h(t)|