 #1
fresh_42
Mentor
 13,240
 10,183
Summary:

functional analysis, operator theory, topology, measure theory, calculus
Authors: Math_QED (MQ), Infrared (IR), Wrobel (WR), fresh_42 (FR).
Main Question or Discussion Point
1. (solved by @nuuskur ) Let ##K## be a nonempty compact subset of ##\Bbb{C}##. Construct a bounded operator ##u: H \to H## on some Hilbert space ##H## that has spectrum ##\sigma(u) =K##. (MQ)
2. Let ##f,g:[0,2]\to\mathbb{R}## be continuous functions such that ##f(0)=g(0)=0## and ##f(2)=g(2)=2##. Show that there exist ##a,b\in [0,2)\,,a\neq b\,.## such that ##f(a)f(b)## and ##g(a)g(b)## are both integers. (IR)
3. (solved by @nuuskur ) Let ##V## be a normed vector space and ##W## be a closed linear subspace. If ##V/W## is a Banach space for the quotient norm, show that ##V## is a Banach space as well. (MQ)
4. (solved by @nuuskur ) Let ##(\Omega, \mathcal{F}, \Bbb{P})## be a probability space and ##X: \Omega \to \Bbb{R}## be a random variable. If ##\Bbb{P}(X \in A) \in \{0,1\}## for all Borel sets ##A##, show that there is a constant ##c \in \Bbb{R}## such that ##\Bbb{P}(X= c) = 1## (i.e. ##X## is constant almost surely). (MQ)
5. Let ##X,Y## be Banach spaces over the field ##\mathbb{R}## or ##\mathbb{C}##. By ##B_X(\hat x,R)## we denote the open ball of this space:
$$B_X(\hat x,R)=\{x\in X\mid \x\hat x\_X<R\},$$ and ## \overline B_X(\hat x,R)=\{x\in X\mid \x\hat x\_X\le R\}.##
Consider a continuous bilinear function ##A:X\times X\to Y## such that
$$\A(x_1,x_2)\_Y\le a\x_1\_X\x_2\_X,\quad A(x_1,x_2)=A(x_2,x_1).$$
Let ##B:X\to Y## stand for a bounded linear operator onto:
$$\overline B_Y(0,b)\subseteq B(\overline B_X(0,1))$$ with some positive constant ##b##. Let ##C## stand for a fixed element of ##Y,\quad \C\_Y=c##.
Show that
If ##D=b^24ac\ge 0## then the equation
$$A(x,x)+Bx+C=0$$ has a solution. (WR)
6. (solved by @nuuskur ) Show that in ZF the Tychonoff theorem (product of compact spaces is compact) implies the axiom of choice. (MQ)
7. Let ##f: (X,d_X) \to (Y,d_Y)## be a map between metric spaces (not necessarily continuous!). Show that ##D(f):= \{x \in X\mid f \mathrm{\ is \ not \ continuous \ at \ x}\}## is a Borelmeasurable set of ##X##. (MQ)
8. (solved by @benorin , @Fred Wright , and @nuuskur ) Let ##F## be a meromorphic function (holomorphic up to isolated poles) in ##\mathbb{C}## with the following properties:
(1) ##F## is holomorphic (complex differentiable) in the half plane ##H(0)=\{z\in \mathbb{C}\, : \,\Re(z)>0\}.##
(2) ##zF(z)=F(z+1).##
(3) ##F## is bounded in the strip ##\{z\in \mathbb{C}\, : \,1\leq\Re(z)\leq 2\}.##
Show that ##F(z)=F(1)\Gamma(z).## (FR)
9. (solved by @PeroK ) Show that there exists no continuous function ##g: \Bbb{C}\setminus \{0\}\to \Bbb{C}## such that ##e^{g(z)} = z## for all ##z\in \Bbb{C}\setminus \{0\}## (i.e. there is no continuous logarithm on ##\Bbb{C}\setminus \{0\})##. (MQ)
(Corrected version, credits go to @PeroK for pointing out and solving the original problem.)
10. (solved by @PeroK ) Show that if ##f## is any continuous real function and ##n## any positive number (FR),
$$
I:=\int_{n^{1}}^{n} f\left(x+\dfrac{1}{x}\right)\,\dfrac{\log x}{x}\,dx =0.
$$
High Schoolers only
11. (solved by @Not anonymous ) Let ##a<b<c<d## be real numbers. Sort ##x=ab+cd\, , \,y=bc+ad\, , \,z=ac+bd## and prove it.
12. (solved by @Not anonymous ) Prove ##\overline{CP}^2=\overline{AP}\cdot \overline{BP}\,.##
13. How big is the probability for two pocket aces in Texas Hold'em? Assume we have seen a show down in a headsup. How many possible combinations are there, how many combinations of possible starting hands can the opponents have? How many possible community cards?
14. (solved by @etotheipi ) Everybody knows that Schrödinger's cat is trapped in the box since ##1935.## Not well known is the fact, that the radioactive material was ten ##{}^{14}C## isotopes. Calculate the probability that the cat is still alive.
15. Show that there is no rational solution for ##p^2+q^2+r^2=7.##
2. Let ##f,g:[0,2]\to\mathbb{R}## be continuous functions such that ##f(0)=g(0)=0## and ##f(2)=g(2)=2##. Show that there exist ##a,b\in [0,2)\,,a\neq b\,.## such that ##f(a)f(b)## and ##g(a)g(b)## are both integers. (IR)
3. (solved by @nuuskur ) Let ##V## be a normed vector space and ##W## be a closed linear subspace. If ##V/W## is a Banach space for the quotient norm, show that ##V## is a Banach space as well. (MQ)
4. (solved by @nuuskur ) Let ##(\Omega, \mathcal{F}, \Bbb{P})## be a probability space and ##X: \Omega \to \Bbb{R}## be a random variable. If ##\Bbb{P}(X \in A) \in \{0,1\}## for all Borel sets ##A##, show that there is a constant ##c \in \Bbb{R}## such that ##\Bbb{P}(X= c) = 1## (i.e. ##X## is constant almost surely). (MQ)
5. Let ##X,Y## be Banach spaces over the field ##\mathbb{R}## or ##\mathbb{C}##. By ##B_X(\hat x,R)## we denote the open ball of this space:
$$B_X(\hat x,R)=\{x\in X\mid \x\hat x\_X<R\},$$ and ## \overline B_X(\hat x,R)=\{x\in X\mid \x\hat x\_X\le R\}.##
Consider a continuous bilinear function ##A:X\times X\to Y## such that
$$\A(x_1,x_2)\_Y\le a\x_1\_X\x_2\_X,\quad A(x_1,x_2)=A(x_2,x_1).$$
Let ##B:X\to Y## stand for a bounded linear operator onto:
$$\overline B_Y(0,b)\subseteq B(\overline B_X(0,1))$$ with some positive constant ##b##. Let ##C## stand for a fixed element of ##Y,\quad \C\_Y=c##.
Show that
If ##D=b^24ac\ge 0## then the equation
$$A(x,x)+Bx+C=0$$ has a solution. (WR)
6. (solved by @nuuskur ) Show that in ZF the Tychonoff theorem (product of compact spaces is compact) implies the axiom of choice. (MQ)
7. Let ##f: (X,d_X) \to (Y,d_Y)## be a map between metric spaces (not necessarily continuous!). Show that ##D(f):= \{x \in X\mid f \mathrm{\ is \ not \ continuous \ at \ x}\}## is a Borelmeasurable set of ##X##. (MQ)
8. (solved by @benorin , @Fred Wright , and @nuuskur ) Let ##F## be a meromorphic function (holomorphic up to isolated poles) in ##\mathbb{C}## with the following properties:
(1) ##F## is holomorphic (complex differentiable) in the half plane ##H(0)=\{z\in \mathbb{C}\, : \,\Re(z)>0\}.##
(2) ##zF(z)=F(z+1).##
(3) ##F## is bounded in the strip ##\{z\in \mathbb{C}\, : \,1\leq\Re(z)\leq 2\}.##
Show that ##F(z)=F(1)\Gamma(z).## (FR)
9. (solved by @PeroK ) Show that there exists no continuous function ##g: \Bbb{C}\setminus \{0\}\to \Bbb{C}## such that ##e^{g(z)} = z## for all ##z\in \Bbb{C}\setminus \{0\}## (i.e. there is no continuous logarithm on ##\Bbb{C}\setminus \{0\})##. (MQ)
(Corrected version, credits go to @PeroK for pointing out and solving the original problem.)
10. (solved by @PeroK ) Show that if ##f## is any continuous real function and ##n## any positive number (FR),
$$
I:=\int_{n^{1}}^{n} f\left(x+\dfrac{1}{x}\right)\,\dfrac{\log x}{x}\,dx =0.
$$
High Schoolers only
11. (solved by @Not anonymous ) Let ##a<b<c<d## be real numbers. Sort ##x=ab+cd\, , \,y=bc+ad\, , \,z=ac+bd## and prove it.
12. (solved by @Not anonymous ) Prove ##\overline{CP}^2=\overline{AP}\cdot \overline{BP}\,.##
13. How big is the probability for two pocket aces in Texas Hold'em? Assume we have seen a show down in a headsup. How many possible combinations are there, how many combinations of possible starting hands can the opponents have? How many possible community cards?
14. (solved by @etotheipi ) Everybody knows that Schrödinger's cat is trapped in the box since ##1935.## Not well known is the fact, that the radioactive material was ten ##{}^{14}C## isotopes. Calculate the probability that the cat is still alive.
15. Show that there is no rational solution for ##p^2+q^2+r^2=7.##
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