Use this tag for questions asking about "problem books", "exercise books", and their solutions.
4
votes
1answer
23 views
$L^{p}$ functions from Rudin Exercises 3.5
I am attempting a question from Rudin's "Real and Complex Analysis" Chapter 3 question 5. I shall summarise the question as below: Suppose that $f$ is a complex measurable function on $X$, $\mu$ a ...
3
votes
1answer
46 views
Simple module is isomorphic to R/M where M is a maximal ideal
In Michael Artin's Algebra textbook page 484 Chapter 12 Exercise 1.6:
A module is called simple if it is not the zero module and if it has no proper submodule.
(a) Prove that any simple module is ...
1
vote
0answers
22 views
Unbounded self- adjoint and von Neumann algebra
I am reading Conway's Functional Analysis. Here is one exercise problem.I don't know how to show the following fact. For unbounded self-adjoint $T$ in Hilbert space $H$
1) $T$ commutes with its Borel ...
1
vote
0answers
19 views
Solvable Lie algebra with codimension 1 ideal
There is an exercise in Humphreys "Any nilpotent Lie algebra contains a codimension 1 ideal".
The proof I am thinking of is the following.
Suppose the Lie algebra $L$ is non-satisfies $L\neq[L,L]$. ...
0
votes
2answers
28 views
Linear Algebra : find the kernel of this transformation.
Q. I think I find the kernel but several... which is correct? Seems like depending on which variable I put as kernel, I can get several kernels. Correct?
T is the transformation from $\mathbb{R}^2$ ...
2
votes
1answer
68 views
question 9 - chap 5 evans PDE
The question is :
Integrate by parts to prove :
$$\int_{U} |Du|^p \ dx \leq C \left(\int_{U} |u|^p \ dx\right)^{1/2} \left(\int_{U} |D^2 u|^p \ dx\right)^{1/2}$$
for $ 2 \leq p < \infty$ ...
1
vote
1answer
45 views
How do I determine a formula for a given trig function?
Assume that 0 < x < pi/2 and sin(x) = z
a.) Find a formula that gives the value of sin(x/2) in terms of z
b.) Corroborate the validity of the formula for these values of x:
pi/4
pi/3
pi/6
...
2
votes
1answer
44 views
Getting an acute angle for an obtuse angle using law of Sines.
I have done this problem over and over again. I even looked up tutorials on how to properly use law of sines. It's rather embarrassing that I'm struggling so much wish this simple trigonometric stuff.
...
6
votes
4answers
93 views
Show that if $T_1$, $T_2$ are normal operators that commutes then $T_1+T_2$ and $T_1T_2$ are normal.
Let $V$ be a finite dimensional inner-product space, and suppose that $T_1$, $T_2$ are normal operators on $V$ that commutes. How to show that $T_1+T_2$ and $T_1T_2$ are then normal?
It is clear if ...
2
votes
1answer
32 views
Abstract integral - Borel measures - $L^p$ spaces
Let $(X,\mu,M)$ be a finite measure space. Suppose $T\colon X \to X$ is measurable and $\mu(T^{-1}E) = 0$ whenever $E \in M$ and $\mu(E)=0$. Prove that these exists $h \in L^1(\mu)$ such that $h ...
1
vote
1answer
31 views
Weak convergence-exercice
Let $\Omega$ be an open set in $\mathbb{R}^n$ and let $(u_n)$ be a bounded sequence in $H^1_0(\Omega).$
Who's the theorem say that we can extract a subsequence denoted $u_{n}$ as $u_n$ weakly ...
4
votes
1answer
71 views
Matrix Norm set #2
As a complement of the question
Matrix Norm set
and in order to complete the Problem 1.4-5 from the book: Numerical Linear Algebra and Optimisaton by Ciarlet. I have this additional conditions:
(3) ...
5
votes
1answer
40 views
Substitution problem
My question is something I've been thinking about for some time now.
Q: Why is it possible to make substitutions or change in variables ?
I mean, how do I know which substitutions are allowed ?
For ...
1
vote
1answer
52 views
Prove there is a Borel measure u such that $u[x,y) = a(y) - a(x)$
If anyone has a solution to the following exercise, I would be grateful. Thanks.
Let $\alpha$ be continuous and increasing on $[a,b]$. Prove that there exists a unique Borel measure $\mu$ on ...
1
vote
1answer
30 views
