 #1
fresh_42
Mentor
 13,141
 9,922
Summary:

Normed, Banach and Hilbert spaces, topology, geometry, linear algebra, integration, flux.
Authors: Math_QED (MQ), Infrared (IR), Wrobel (WR), fresh_42 (FR).
Main Question or Discussion Point
Questions
1. (solved by @nuuskur ) Let ##H_1, H_2## be Hilbert spaces and ##T: H_1 \to H_2## a linear map. Suppose that there is a linear map ##S: H_2 \to H_1## such that for all ##x\in H_2## and all ##y \in H_1## we have
$$\langle Sx,y \rangle = \langle x, Ty \rangle$$
Show that ##T## is continuous. (MQ)
2. (solved by @strangerep , @PeroK ) Let ##A## and ##B## be complex ##n\times n## matrices such that ##ABBA## is a linear combination of ##A## and ##B##. Show that ##A## and ##B## must have a common eigenvector. (IR)
3. (solved by @zinq ) Give an example of a vector space ##X## over ##\mathbb{R}## and a convex subset ##Q\subset X,\quad Q\ne X## such that ##Q## is not contained in any halfspace of ##X##. (WR)
4. (solved by @benorin , @Delta2 ) Let ##S:=\{\,(x,y,z)\in \mathbb{R}^3\,\,x^2+y^2=(2z)^2,\,0\leq z\leq 2\,\}## be the surface of a cone ##C## with a circular cross section and a peak at ##(0,0,2)##. The orientation of ##S## be such, that the normal vectors point outwards. Calculate the flux through ##S## of the vector field (FR)
$$
F\, : \,\mathbb{R}^3 \longrightarrow \mathbb{R}^3\, , \,F(x,y,z)=\begin{pmatrix}xy^2\\x^2y\\(x^2+y^2)(1z)\end{pmatrix}.
$$
5. (solved by @nuuskur ) Let ##X## be a normed space and ##f:X\to\mathbb{R}## be a linear functional that is not bounded. Show that ##\ker f## is dense in ##X##. (WR)
6.a.) Let ##G## be a group with a compact Hausdorff topology for which multiplication is continuous. Show that ##G## is a topological group, i.e. show that inversion is continuous. (MQ)
6.b.) Is this still true if we don't assume that ##G## is compact Hausdorff? If yes, give a proof. Otherwise, present a counterexample. (MQ)
7. (solved by @StoneTemplePython ) Let ##A## be an ##n\times n## matrix such that
8. (solved by @Fred Wright , @benorin ) Calculate for ##\alpha\geq 1##
$$
\int_0^\pi \log (12\alpha \cos(x)+\alpha^2)\,dx
$$
a.) without using series expansions
b.) by using serious expansions. (FR)
9. Let ##X## be a Banach space and let ##Y## be a normed space. There is a linear operator ##A:X\to Y##.
As usual ##X',Y'## stand for the spaces of bounded linear functionals and the dual operator ##A':Y'\to X'## is defined. Perhaps ##A'## is not bounded but it is defined on the whole ##Y'##.
Prove that ##A## is bounded. (WR)
10. (solved by @benorin ) Evaluate ##\int_0^{\infty}\frac{\log(x)}{x^21}dx.## (IR)
High Schoolers only
11. (solved by @Halc , @etotheipi ) You have two nails in a wall. Can you hang a rope around them in such a way that if either nail is removed, then the rope will fall to the floor?
12.a.) (solved by @etotheipi , @Adesh ) Show that ##\sqrt{i^i} \in \mathbb{R}## where ## i ## is the imaginary unit ##i=\sqrt{1}.##
12.b.) (solved by @Adesh ) Which of the following equation signs is wrong and why?
$$
1 \stackrel{(1)}{=}i\cdot i \stackrel{(2)}{=}\sqrt{1}\cdot \sqrt{1}\stackrel{(3)}{=}\sqrt{(1)\cdot(1)}\stackrel{(4)}{=}\sqrt{1}\stackrel{(5)}{=}1
$$
12.c.) Calculate all solutions of ##z^3=1## by three different methods.
13. (solved by @Adesh , @PhysicsBoi1908 ) Which is the smallest natural number ##n\in \mathbb{N}_0## such that there are no integers ##a,b\in \mathbb{Z}## with ##3a^3+b^3=n?##
14. (solved by @etotheipi ) Is it possible to cover an equilateral triangle with two smaller equilateral triangles without a gap? It's not required that they are of equal area, nor that they won't overlap, only that they are smaller and together have a greater area than the original triangle.
15. (solved by @PhysicsBoi1908 ) Prove: Given ##n## different integers ##\{\,a_1,\ldots,a_n\,\}##, then there exists a subset ##\{\,a_{j_1},\ldots,a_{j_m}\,\}## with ##1\leq j_1 < \ldots < j_m \leq n## such that ##n## divides ##a_{j_1}+\ldots+a_{j_m}\,.##
1. (solved by @nuuskur ) Let ##H_1, H_2## be Hilbert spaces and ##T: H_1 \to H_2## a linear map. Suppose that there is a linear map ##S: H_2 \to H_1## such that for all ##x\in H_2## and all ##y \in H_1## we have
$$\langle Sx,y \rangle = \langle x, Ty \rangle$$
Show that ##T## is continuous. (MQ)
2. (solved by @strangerep , @PeroK ) Let ##A## and ##B## be complex ##n\times n## matrices such that ##ABBA## is a linear combination of ##A## and ##B##. Show that ##A## and ##B## must have a common eigenvector. (IR)
3. (solved by @zinq ) Give an example of a vector space ##X## over ##\mathbb{R}## and a convex subset ##Q\subset X,\quad Q\ne X## such that ##Q## is not contained in any halfspace of ##X##. (WR)
4. (solved by @benorin , @Delta2 ) Let ##S:=\{\,(x,y,z)\in \mathbb{R}^3\,\,x^2+y^2=(2z)^2,\,0\leq z\leq 2\,\}## be the surface of a cone ##C## with a circular cross section and a peak at ##(0,0,2)##. The orientation of ##S## be such, that the normal vectors point outwards. Calculate the flux through ##S## of the vector field (FR)
$$
F\, : \,\mathbb{R}^3 \longrightarrow \mathbb{R}^3\, , \,F(x,y,z)=\begin{pmatrix}xy^2\\x^2y\\(x^2+y^2)(1z)\end{pmatrix}.
$$
5. (solved by @nuuskur ) Let ##X## be a normed space and ##f:X\to\mathbb{R}## be a linear functional that is not bounded. Show that ##\ker f## is dense in ##X##. (WR)
6.a.) Let ##G## be a group with a compact Hausdorff topology for which multiplication is continuous. Show that ##G## is a topological group, i.e. show that inversion is continuous. (MQ)
6.b.) Is this still true if we don't assume that ##G## is compact Hausdorff? If yes, give a proof. Otherwise, present a counterexample. (MQ)
7. (solved by @StoneTemplePython ) Let ##A## be an ##n\times n## matrix such that
 the offdiagonal entries of ##A## are all positive
 the sum of the entries in each row is negative.
8. (solved by @Fred Wright , @benorin ) Calculate for ##\alpha\geq 1##
$$
\int_0^\pi \log (12\alpha \cos(x)+\alpha^2)\,dx
$$
a.) without using series expansions
b.) by using serious expansions. (FR)
9. Let ##X## be a Banach space and let ##Y## be a normed space. There is a linear operator ##A:X\to Y##.
As usual ##X',Y'## stand for the spaces of bounded linear functionals and the dual operator ##A':Y'\to X'## is defined. Perhaps ##A'## is not bounded but it is defined on the whole ##Y'##.
Prove that ##A## is bounded. (WR)
10. (solved by @benorin ) Evaluate ##\int_0^{\infty}\frac{\log(x)}{x^21}dx.## (IR)
High Schoolers only
11. (solved by @Halc , @etotheipi ) You have two nails in a wall. Can you hang a rope around them in such a way that if either nail is removed, then the rope will fall to the floor?
12.a.) (solved by @etotheipi , @Adesh ) Show that ##\sqrt{i^i} \in \mathbb{R}## where ## i ## is the imaginary unit ##i=\sqrt{1}.##
12.b.) (solved by @Adesh ) Which of the following equation signs is wrong and why?
$$
1 \stackrel{(1)}{=}i\cdot i \stackrel{(2)}{=}\sqrt{1}\cdot \sqrt{1}\stackrel{(3)}{=}\sqrt{(1)\cdot(1)}\stackrel{(4)}{=}\sqrt{1}\stackrel{(5)}{=}1
$$
12.c.) Calculate all solutions of ##z^3=1## by three different methods.
13. (solved by @Adesh , @PhysicsBoi1908 ) Which is the smallest natural number ##n\in \mathbb{N}_0## such that there are no integers ##a,b\in \mathbb{Z}## with ##3a^3+b^3=n?##
14. (solved by @etotheipi ) Is it possible to cover an equilateral triangle with two smaller equilateral triangles without a gap? It's not required that they are of equal area, nor that they won't overlap, only that they are smaller and together have a greater area than the original triangle.
15. (solved by @PhysicsBoi1908 ) Prove: Given ##n## different integers ##\{\,a_1,\ldots,a_n\,\}##, then there exists a subset ##\{\,a_{j_1},\ldots,a_{j_m}\,\}## with ##1\leq j_1 < \ldots < j_m \leq n## such that ##n## divides ##a_{j_1}+\ldots+a_{j_m}\,.##
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