0
votes
1answer
19 views

Understanding why $\int_\gamma {dz \over z - a} = k 2\pi i$ for $\gamma$ a closed curve not passing through $a$

The following is a paraphrased proof from Ahlfors. I bolded the part that is confusing me and asked a question about it at the bottom of this post. Hypothesis: Let $\gamma$ be a closed curve that ...
3
votes
2answers
24 views

Analytic $F(z)$ has $f(z)$ as derivative $\implies$ $\int_\gamma f(z)\ dz = 0$ for $\gamma$ a closed curve

Hypothesis: Suppose that $F(z)$ has $f(z)$ as a derivative. Suppose further that $F(z)$ is analytic. Now consider the complex line integral $$ \tag{1} \int_\gamma f(z)\ dz $$ Question: Does this ...
0
votes
1answer
23 views

Verifying a condition for which $\int_\gamma p\ dx + q\ dy$ depends only on endpoints

Hypothesis: Suppose there exists a function $U(x,y)$ in $\Omega$ with partial derivatives $${\partial U \over \partial x} = p \quad \quad {\partial U \over \partial y} = q$$ Goal: Show that the ...
1
vote
0answers
17 views

Computing a complex line integral $dz$ in terms of line integrals $dx$ and $dy$

Goal: I'm trying to verify the calculation claimed by Ahlfors that $$\int_\gamma f(z)\ dz = \int_\gamma (u\ dx - v\ dy) + i \int_\gamma (u\ dy + v\ dx)$$ Attempt: $$\int_\gamma (u\ dx - v\ dy) + i ...
2
votes
1answer
18 views

Some questions about proof of Theorem 2.43 in Baby Rudin

I will include the proof here and highlight the parts that are giving me trouble. Theorem $\hspace{5 pt}$ Let $P$ be a nonempty perfect set in $\mathbb{R}^k$. Then $P$ is uncountable. Proof ...
1
vote
0answers
21 views

A simple proof with directional derivatives

Suppose $\nabla f (x) \cdot d < 0$, prove that there exists $\delta > 0$ such that $$f(x + \tau d ) < f(x)$$ for all $\tau \in (0, \delta)$ My proof consists of only a few lines. ...
0
votes
2answers
29 views

If f is continuos on an interval, is it then uniformly continuous

I have a function, $f$ that is differentiable on $(0,5)$, and I know it is continuous on $(0,5).$ Is it also uniformly continuous on $(0,5)?$ I now know that it is not. Can someone give me a proof ...
0
votes
1answer
24 views

If $f$ is continuous on $(0,5)$, is it uniformly continuous on same interval

I have a function, $f$ that is differentiable on $(0,5)$, and I know it is continuous on $(0,5).$ Is it also uniformly continuous on $(0,5)?$ I believe it is. I now know that it is not. Can someone ...
0
votes
1answer
15 views

Holder's Inequality Proof Verification

Wikipedia outlines a nice proof of Holder's Inequality in the link provided. The fifth sentence in the proof reads: Dividing $f$  and $g$ by $\|f \|_p$ and $\|g\|_q$, respectively, we can assume ...
2
votes
0answers
26 views

Prove that $\lim\limits_{t \rightarrow 0} \int\limits_{0}^{\infty} e^{-tx} f(x) dx = 1$ for $t>0$.

Suppose $f \in \mathcal{R}$ on $[0,A]$ for all $A < \infty$, and $f(x) \rightarrow 1$ as $x \rightarrow + \infty$. Prove that $\lim\limits_{t \rightarrow 0} \int\limits_{0}^{\infty} e^{-tx} f(x) ...
2
votes
1answer
110 views

Proof concerning equivalent definition of supremum using limits

I believe I have a proof to the following theorem: Let $S$ be a subset of $\mathbb{R}$ that is non-empty and bounded above. $s \in \mathbb{R}$ is the supremum iff $s$ is an upper bound of $S$ and for ...
1
vote
1answer
21 views

Verify why this is not a metric $d(x,y)=\|x-y\|_p$ ( $\|x\|_p=p^{-h}$ if $x=p^h\dfrac{m}{n}$).

$d(x,y)=\|x-y\|_p$ $p$ is prime and $\|x\|_p=p^{-h}$ if $x=p^h\dfrac{m}{n}$, where $m, n$ are coprimes with $p$. This is not a metric because if $x=y=p^k\dfrac{m}{n}$, then $x-y=0=p^0\dfrac0n$. ...
1
vote
1answer
28 views

$v$ is Conjugate Harmonic to $u$ $\implies$ $f = u + iv$ is Analytic (Proof Verification from Ahlfors)

Hypothesis: Let $u$ and $v$ be two functions from $\mathbb{R}^2$ to $\mathbb{R}$ s.t. $$ \Delta u = {\partial^2 u \over \partial x^2} + {\partial^2 u \over \partial y^2} = 0 $$ and $$ \Delta v = ...
2
votes
4answers
63 views

A question about metrizability

In a lecture in Topology I had earlier this week, I was told (without proof) that not every topological space $(X,O)$ is metrizable, i.e, it is impossible to find some metric $d$ such that $O$ and ...
4
votes
2answers
35 views

Orthogonal representation of finite operator

I would like to know if my proof is correct. Statement: Let $T$ be a finite rank operator on a Hilbert space $\mathscr{H}$. Show that $\forall \, h \, \in \mathscr{H}, \, T(h)$ can be written as ...
5
votes
0answers
80 views

$f$ continuous nowhere implies $f$ not Riemann integrable

I am trying to prove the statement in the title, without using the theorem which says that the set of discontinuities of a Riemann integrable function must have measure $0$. I am aware that similar ...
0
votes
1answer
44 views

Isometry is not surjective

According to the definition I am using, an isometry is a mapping $f:X \rightarrow Y$ between two metric spaces $(X,d_{X})$ and $(Y,d_{Y})$: $$ d_{Y}(f(a),f(b)) = d_{X}(a,b) $$ for all $a,b \in X $ I ...
1
vote
1answer
62 views

Possible book correction or am I missing something?

Hi I am teaching myself analysis and bought "Analysis - With an introduction to Proof" by Steven R. Lay. Now one of the practice problems is "Determine the truth value of each statement, assuming x, y ...
0
votes
1answer
36 views

Showing that $\log \log(z)$ is Analytic (Proof Verification)

Goal: Convert $\log \log (z)$ into a single-valued function defined on a suitable region of $\mathbb{C}$ and then prove that it is analytic. Attempt: As has been demonstrated elsewhere, we have ...
0
votes
0answers
49 views

Showing that $\sqrt{1+z} + \sqrt{1-z}$ is Analytic (Proof Verification)

Ahlfors: Give a precise definition of a single-valued branch of the function $\sqrt{1+z} + \sqrt{1-z}$ in a suitable region, and prove that it is analytic. Is my following proof attempt valid? ...
1
vote
2answers
61 views

$\forall x \in \mathbb{R}$ show that $x=\sum_{n=1}^\infty k_na_n = \prod_{n=1}^{\infty}m_na_n$ …

Yet again, another cool problem from the book "problems in mathematical analysis" by Piotr & Witkowski: Prove that if $a_n \neq 0$, $n=1,2,\cdots$ and $\displaystyle \lim_{n \to \infty} a_n = 0$, ...
1
vote
0answers
31 views

If $(\sqrt{5}-1)/2 = \sum_{k=1}^{\infty}2^{-n_k}$ where $n_k \in \mathbb{N}$ then $n_k \leq 5\cdot 2^{k-1}-1$

Show that if $(\sqrt{5}-1)/2 = \sum_{k=1}^{\infty}2^{-n_k}$ where $n_k$ are positive integers, then $n_k \leq 5\cdot 2^{k-1}-1$. This is a problem from the book "Problems in mathematicaly analysis" ...
0
votes
0answers
20 views

Proof Check: Closed range then bounded below

Statement: Given a Hilbert $\mathscr{H}$, and $T \in \mathscr{B}(\mathscr{H}, \mathscr{H})$, where $T$ has closed range. Prove that for all $h \in N(T)^\perp$ then $\exists \, m>0 \, \mbox{s.t.} \, ...
0
votes
1answer
41 views

Proof check for $(X/M)^{*} \cong M^{\perp}$

I would like to know if the proof I have is correct. Statement: Let $M$ be a closed subspace if a Banach space $X$. Let $\pi: X \rightarrow X/M$ be the quotient map. Put $Y= X/M$ for each $\varphi \, ...
0
votes
1answer
27 views

Showing the radius of convergence of $\sum a_n b_n z^n$ is at least $R_1 R_2$

Problem: If $\sum a_n z^n$ and $\sum b_n z^n$ have radii of convergence $R_1$ and $R_2$, show that the radius of convergence of $\sum a_n b_n z^n$ is at least $R_1 R_2$. Is the following proof ...
0
votes
1answer
18 views

Finding the radius of convergence for $\sum n^p z^n$ (Proof Verification)

Goal: Find the radius of convergance for the following complex power series: $$ \sum n^p z^n $$ Attempt: We have by Hadamard's formula for the radius of convergence that the complex power series ...
2
votes
1answer
32 views

Harmonic map into sphere

Let $B$ be the unit ball and $S$ the unit sphere in $\mathbb{R}^3$. Consider the map $u: B\rightarrow S$ defined as: \begin{equation} u^j(x)=\frac{x_j}{|x|}\quad\forall \ j =1, 2, 3. \end{equation}I ...
0
votes
0answers
20 views

Proof similar to that of Baire's theorem

Sometimes, I read proofs that are very similar to the proof of Baire's theorem, but which I can't directly simplify by using Baire's theorem. I'll give an example: Let $M$ be a complete metric or ...
2
votes
0answers
53 views

limit proof at a point. Spivak, Chapter 5 problem 36b.

Prove that $\lim_{x\to0^-} f(\frac{1}{x})=L$ iff $\lim_{x\to-\infty} f(x)=L$ Proof $(\leftarrow)$ Suppose that $\lim_{x\to-\infty} f(x)=L$ Let $\epsilon>0$ then there exists a real number ...
1
vote
2answers
40 views

$\pi \cot (\pi z) = \frac{1}{z} + \sum_{n \in \mathbb{Z},\ n \neq 0} \frac{1}{z-n}+ \frac{1}{n}$

I'm reading the proof that $$\pi \cot (\pi z) = \frac{1}{z} + \sum_{n \in \mathbb{Z},\ n \neq 0} \frac{1}{z-n}+ \frac{1}{n}$$ There is a function $$h(z) =\pi \cot (\pi z) -[ \frac{1}{z} + \sum_{n ...
1
vote
2answers
115 views

Showing that $A=B+\alpha \cdot I$ is an invertible matrix

Let $B$ be a non-zero random $n\times n$ matrix generated using the matlab command $B=rand(n,n)$. I need to show that $A=B+\alpha \cdot I$ is an invertible matrix, where $\alpha=\|B\|_{\infty}$. I ...
3
votes
2answers
87 views

Why is the set of all $\infty$-tuples with finitely many non-zero rational terms dense in $\ell_2$?

This statement has been given as an example in the book "Introductory real analysis" written by Kolmogorov and Fomin: The set of all points $x=(x_1,x_2,\cdots,x_n,\cdots)$ with only finitely ...
1
vote
1answer
45 views

A rearrangement of an absolutely convergent complex series is also absolutely convergent

I just completed the following proof. Is it valid? Let $\sum_{k=1}^{\infty} a_k$ be an arbitrary convergent series that also converges absolutely. Then $\sum_{k=1}^{\infty} a_k \in \mathbb{C}$ and ...
0
votes
0answers
45 views

$f \in \mathcal{R}(\alpha)$ on $[a,b]$, then $\exists P_n$ s.t. $\lim\limits_{n \rightarrow \infty} \int\limits_{a}^{b} |f-P_n|^2 d \alpha =0$.

Assume $f \in \mathcal{R}(\alpha)$ on $[a,b]$, and prove that there are polynomial $P_n$ such that $\lim\limits_{n \rightarrow \infty} \int\limits_{a}^{b} |f-P_n|^2 d \alpha =0$. This is what I have, ...
3
votes
1answer
113 views

If $f$ is continuous on $[0,1]$ and if $\int\limits_{0}^{1} f(x) x^n dx = 0, (n=0,1,2,…)$, prove that $f(x)=0$ on $[0,1]$.

If $f$ is continuous on $[0,1]$ and if $\int\limits_{0}^{1} f(x) x^n dx = 0, (n=0,1,2,...)$, prove that $f(x)=0$ on $[0,1]$. This is what I have, how does it look? Proof: Let $P(x)$ be any ...
0
votes
0answers
86 views

$\{f_n\}$ equicontinuous sequence of functions on compact $K$, converges pointwise on $K$ then converges uniformly on $K$.

Suppose $\{f_n\}$ is an equicontinuous sequence of functions on a compact set $K$ and $\{f_n\}_{n=1}^{\infty}$ converges pointwise on $K$. Prove that $\{f_n\}_{n=1}^{\infty}$ converges uniformly on ...
1
vote
0answers
45 views

Prove that every compact metric space $K$ has a countable base, and that $K$ is therefore separable.

Prove that every compact metric space $K$ has a countable base, and that $K$ is therefore separable. How does the following look? Proof: For each $n \in \mathbb{N}$, make an open cover of $K$ by ...
3
votes
0answers
73 views

Prove that every separable metric space has a countable base.

A collection $\{V_\alpha \}$ of open subsets of $X$ is said to be a base for $X$ if the following is true: For every $x \in X$ and every open set $G \subset X$ such that $x \in G$, we have $x \in ...
4
votes
0answers
70 views

Prove that $\mathbb{R}^k$ is separable

I'd like to show that $\mathbb{R}^k$ is separable. (A metric space is called separable if it contains a countable dense subset.) Here's what I have and I'd like to confirm with everyone to see if ...
3
votes
2answers
96 views

Question about sup norm

Let $x \in \mathbb{R}^n$. Define $|x| = \max\{ |x_1|,...,|x_n|\} $. I want to show that this is a norm on $R^n$. This is my reasoning. First, notice $$ |x| = \max\{ |x_i| \} \geq |x_i| \; \forall i ...
1
vote
1answer
56 views

Interior ball condition

Let $\Omega\subset\mathbb{R}^n$ be a open set. We say that $y\in\partial \Omega$ satisfies the interior ball condition, if there is $x\in \Omega$ and $r>0$ such that $$B(x,r)\subset\Omega,\ y\in ...
1
vote
1answer
74 views

If $X$ is $G$-paradoxical then $G$ is $G$-paradoxical. Is my proof correct?

I am currently reading Stan Wagon's Banach-Tarski Paradox book, and this was left as an exercise to prove (converse of Proposition 1.10). Let $X$ be a set, and let $G$ act on $X$ with no ...
6
votes
0answers
107 views

Fixed point of a shrinking map. Proof.

Could you please verify my proof? Let $f:X \to X$ be a shrinking map on a compact metric space $(X,d)$. In other words: $d(f(x),f(y)) < d(x,y)$ for all $x,y \in X$. Prove: $f$ has a unique ...
3
votes
0answers
64 views

For the sequence $u_n$, $u_n \to +\infty \iff \frac{1}{u_n} \to 0$

Let $u=(u_n)_{n \in \mathbb{N}}$ be a sequence such that $u_n \neq 0$, $u_n \to +\infty$, for $ n \to +\infty$. Proof that $u_n \to + \infty , ( n \to +\infty) \iff \left(( \exists n_0 , ...
1
vote
2answers
102 views

Trying to prove that the sequence: 3, 3, sin(1), 3, 3, sin(2), 3, 3, sin(3), 3, 3, sin(4), $\ldots$ does not converge to 3

$\textbf{Proof}$: I need to show that $\exists \epsilon > 0$, such that, $\forall N \in \mathbb{N},\ \exists n \ge N$, such that, $\ \left | a_n - 3 \right | \ge \epsilon$ Let $\epsilon = 2$, ...
1
vote
1answer
34 views

I am trying to prove that for all $x \ge 1, (x^{\frac{1}{n}}) \to 1$.

$\textbf{Proof:}$ Let $x \ge 1$ be arbitrary and note that $\forall n \in \mathbb{N}$, $x^{\frac{1}{n}} > 1$. So $ \forall n \in \mathbb{N},\ x^{\frac{1}{n}} - 1 \ge 0 > -1$. Therefore I ...
0
votes
1answer
26 views

Need help clarifying a proof ( limSn=SupS)

Let $S$ be a bounded nonempty subset of $R$ such that $Sup(S)$ is not in $S$. Prove $\exists$ a sequence $(S_n)$ of points that belong to $S$ such that $ limS_n=Sup(S)$. Let $t=Sup(S)$.then for ...
0
votes
0answers
47 views

Is this integral a counter example to this theorem?

I may have misunderstood the proposition, but I thought it was: Let $f$ be a function $[a,b]\times I\to \Bbb R$, where $I$ is some real interval. Then a sufficient condition for ...
2
votes
1answer
169 views

Sufficient and necessary conditions to get an infinite fiber $g^{-1}(w)$

I want to verify the proof of this result and get some start ideas to overcome the different steps of this proof. Lemma: Let $g$ be a real analytic function. Then we have the equivalence ...
1
vote
0answers
52 views

How to prove and what are the necessary hypothesis to prove that $\frac{f(x+te_i)-f(x)}{t}\to\frac{\partial f}{\partial x_i}(x)$ uniformly?

Let $U\subset\mathbb{R}^n$ be a open set and $f:U\to\mathbb{R}$ a function in $C^\infty_c(U)$. Evans PDE book uses the following result $$\frac{f(x+te_i)-f(x)}{t}\to\frac{\partial f}{\partial ...