# In a thesis, is it appropriate to allude to results obtained later on?

I want to provide a result in chapter 2, but formally prove this result in chapter 8.Can I say something like, "as we will see in chapter 8, this result in chapter 2 is true"?

I want to provide a result in chapter 2, but formally prove this result in chapter 8.Can I say something like, "as we will see in chapter 8, this result in chapter 2 is true"?

It’s an exercise in frustration for

everyone involves, and it’s time to get off the treadmill.”

- Lost At School, Chapter 01

Is "get off the treadmill" a slang?

I have purchased an InDesign template, complete with text, headings, and chapter numbers.I have deleted chapters 2 - 8, and now I want to rename the text Chapter IX to Chapter II, but the text is not editable - it looks like it is a predefined section.

I am working through the Augmented Sixth Chords chapter of Piston's Harmony (5th Edition).The last figured bass exercise, 1g ends with the following cadence.

I've been reading about Moshiach for about a year.Then the other day, I read the above 9 lines (Proverbs 8:22-31).

In Structure and Geometry of Lie Groups by Hilgert and Neeb there is an exercise to construct a connected subgroup of $(\mathbb{R}/\mathbb{Z})^2$ which is not path connected (Exercise 9.Considering this is an exercise in a basic book on Lie groups (appearing in the first chapter on them!

Chapter $3$ of Functional Analysis, Sobolev Spaces and Partial Differential Equations by Haim Brezis constructs and explains the Weak and Weak$^*$ topologies over a Banach Space $E$.The most remarkable result of these topologies is the Banach-Alaoglu-Bourbaki, which asserts that for the weak$^*$ topology (Over $E^*$), the unit ball: $$B_{E^*}=\{f \in E^* | \ ||f|| \leq 1 \}$$ is compact.

This is exercise 38 from Chapter 3.Modules and Vector Spaces in Algebra by Adkins and Weintraub (GTM).

In Rudin's Functional Analysis

Chapter 4 exercise 21, the last formula:

$$\textrm{Re}\, \sum_{n=1}^\infty \alpha_n \langle x_n, x^* \rangle \leq 1 $$

Shouldn't the inequality be reversed?The whole exercise:

Length of curve $(t, \log t)$ from $1$ to $2$.This problem is from Lang's Calculus of Several Variables; chapter 2, subsection 2, exercise 5.

This the exercise 10 of chapter 2 of Bourbaki "Théorie des ensembles", whose intention is to generalize the correspondence between equivalence relations and partitions.3) For distinct elements $a$ and $b$ of $X$ such that $aRb$, we define $C(a,b)=\{x\in X:(aRx\wedge bRx)\}$.

I am still working on Spivak calculus and at problem 6 of the third chapter, I came across an exercise that has been confusing me for a couple hours.I understand the solution (well, I understand how I got there but I have no idea what my solution represents).

I am trying to solve exercise 5, Chapter 3 of Rudin.Hint: By definition, $\limsup (a_n)$ is the infimum of the numbers $s_k = \sup \{a_k, a_{k+1},.

In Exercises 1 to 8 a formula is given for a function $F:\mathbb{R}^2\to\mathbb{R}^2$.In each exercise determine if $F$ is linear.

It is from the book "linear algebra done right" 2nd edition, chapter 3, exercise 4.Prove that if u ∈ V is not

in null T, then V = null T ⊕ {au : a ∈ F}.

But the solution to this exercise goes as follows:

"$f'(x)= g'(t+x)$ by problem 8(a), $f'(t)= g'(t+x)$ again by problem 8(a).Hence $f'(x)= g'(2x).

Show $X_n\Rightarrow X_0$, if and only iff, for every $k\geq 0$, $P[X_n=k]\rightarrow P[X_0=k]$.My work:

$P[X_n=k]\rightarrow P[X_0=k]$ implies $X_n\Rightarrow X_0$ due to Sheffe's Lemma (convergence of densities implies weak convergence).

I am attempting to self-study through Baby Rudin and I have done every exercise in chapter 1 except for problem 16.Every solution I find on the internet for this problem uses theorems from linear algebra which I have no background in.

Let $V$ be the space of $2\times 2$ matrices over $F.$ Find a basis $\{A_1,A_2,A_3,A_4,\}$ for $V$ such that $A_j^2=A_j,$ for each $j.

Determine

$$N^\bot=\{x\in E : \langle f, x\rangle = 0 \quad \forall f\in N \}$$

and

$$N^{\bot\bot}=\{f\in E^* : \langle f, x\rangle = 0 \quad \forall x\in N^\bot \}.$$

Check that $N^{\bot\bot} \ne N^\bot$?