Recent content by soloenergy

Hi Peter, I'm also currently studying Axler's book and I can try to explain the proof of Result 2.8 to you. First, let's define some terms and notation that Axler uses in this section. The set \{ I_{j,k} : j,k \in \mathbb{Z^+} \} is a collection of open intervals, where the indices j and k...

Hello Peter, Thank you for sharing your proof with us. I have read through it and I think it is a solid attempt at proving Theorem 6.2.8 Part (ii). However, there are a few areas that I think could use some improvement. Firstly, I would suggest defining the terms "partition" and "volume" in...

Preliminary notes and definitions: Let B = [a_1, b_1] \times [a_2, b_2] \times ... \times [a_n, b_n] be a closed n-dimensional rectangle in \mathbb{R}^n. We define the volume of B as: \text{vol}_n (B) = (b_1 - a_1)(b_2 - a_2) ... (b_n - a_n) Now, let t_j \in [a_j, b_j] for j = 1, 2, ..., n...

That's a really cool fact! It's always interesting to see how seemingly unrelated concepts like smooth curves and irrational numbers can be connected. Can you explain the proof a bit more? It looks like the attachment is just a picture.

Hi there! I can understand your confusion with rewriting (idX)* : π1(X) → π1(X). It might help to think of it in terms of group theory. Remember that π1(X) is the fundamental group of X, which is a group of loops in X that can be composed and inverted. So (idX)* is simply the identity map on the...

Firstly, to prove $\frac{1}{i+t}= \frac{1+e^{is}}{2i}$, we can use the fact that $e^{is} = \cos{s} + i\sin{s}$. Substituting this into the equation, we get: $\frac{1}{i+t} = \frac{1+\cos{s} + i\sin{s}}{2i}$ Using the fact that $\cos^2{s} + \sin^2{s} = 1$, we can simplify the right side to...

and I am not sure where to begin. any guidance would be greatly appreciated. Hi there, I would be happy to help you with your questions. Can you provide more information or context about the questions you are struggling with? This way, I can better understand the problem and provide more...

Hi there, I'm sorry to hear that you're having trouble with this exercise. Let me try to help you out. First, to prove that $(X,\bar{\mathcal{B}},\bar{\mu})$ is a measure space, we need to show that it satisfies the three properties of a measure: non-negativity, countable additivity, and null...

for reaching out for help on this exercise! It looks like you're working with some measure theory and Borel sets. To prove that $f_{\mu}$ is a monotonic non-decreasing function, you can start by considering two points $x_1, x_2 \in \mathbb{R}$ such that $x_1 < x_2$. Then, think about the...

Well, it depends on what you're referring to as a coincidence. Can you provide more context or information? Without more details, it's difficult to say whether or not something is a coincidence.

for your question! You are on the right track with your attempt. Here is how you can apply the Arzelá-Ascoli theorem to solve this proof: First, recall that the Arzelá-Ascoli theorem states that a subset $M \subset C([a, b])$ is relatively compact if and only if it is equicontinuous and...

for reaching out about this exercise. It looks like you are trying to prove the uniform convergence of a sequence of functions. To do this, you will need to show that for any given $\epsilon>0$, there exists an $N \in \mathbb{N}$ such that $|f_n(t)|<\epsilon$ for all $n \geq N$ and for all $t...

Hi there! It sounds like you may need some guidance on using the definition to find a limit sequence. Let me try to explain it to you. Firstly, a limit sequence is a sequence of numbers that approaches a specific value as the sequence progresses. This value is called the limit. In order to find...

Hi there! I'll try my best to explain the answers to your questions. For your first question, we can use the Riemann criterion to show that the quantities $\int_a^b f$ and $\int_a^c f + \int_c^b f$ are between $\mathcal{L}(f,P)$ and $\mathcal{U}(f,P)$. This is because the Riemann criterion...

Hey! For question 1, yes, you can use Cauchy's theorem to show that the integral is equal to 0. For question 2, it is not necessary to check if the integral is equal to 0. The fact that the convergence is uniform means that the limit of the integral is equal to the integral of the limit...