Mathematical analysis. Consider a more specific tag instead: (real-analysis), (complex-analysis), (functional-analysis), (fourier-analysis), (measure-theory), (calculus-of-variations), etc. For data analysis, use (data-analysis).

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1answer
17 views

Inequality for $u \in L^r(\Omega)$: $\int_{\Omega} (a_1+a_2|u|^{r/s})^s dx \leq a_3 \int_{\Omega} (1+|u|^r)dx$

This question is from one of the steps of the Proof of Proposition B.1 in Appendix B of P. H. Rabinowitz's "Minimax Methods in Critical Point Theory with Applications to Differential Equations." Let ...
1
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0answers
19 views

Fourier coefficients of the Gaussian.

I would need to find the fourier coefficient of this gaussian for a problem. I'm now stuck with this integral, \begin{equation} c_{n}=\int_{-1}^{1}e^{\frac{x^{2}}{2}}\left(\cos\left(\pi ...
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4answers
46 views

limit of form “$∞ \cdot 0$”

I am trying to formally prove that limit of $2^n\sin(π/2^n)$ as $n$ approaches infinity is $π$. Generally I can tell limit of each term of product of $∞$ and $0$ respectively, but am little confused ...
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1answer
16 views

question about weierstrass approximation theorem true or false justify [on hold]

Is the following assertion true or false? There exists a nonzero function $f \in C([0,1])$ such that $$\int_0^1f(x)x^ndx=0 (\forall n \in \mathbb N)$$ holds. (Hint: use the weierstrass approximation ...
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0answers
8 views

When is the Stieltjes integral of bounded variations?

I was trying to figure out when a Riemann or Lebsgue Stieltjes integral is of bounded variation. For simplicity let $f$ be a increasing RCLL function; when is that $$\int_0^t g(x) df(x)$$ is of ...
1
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1answer
15 views

Application of uniform boundedness principle for sequence of operators

Let $T_n:X:=C([0,1])\rightarrow C([0,1])=:Y$ be a sequence of linear operators. Suppose: $T_n$ is bounded $(Tf)(x):=\lim\limits_{n\to\infty}(T_nf)(x)$ exist for every $f\in C([0,1])$ and $x\in ...
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1answer
25 views

The derivatives of Riemann xi function

What are the first few values of derivatives of Riemann xi function at zero? Is there any general formula for calculating the nth derivative of the riemann zeta function at zero? What happens to the ...
0
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1answer
34 views

Why is continuity needed to substitute value of derivative inside Riemann-Stieltjes Integral?

Given $f$ increasing on $[a,b]$, $g(x)\in R(\alpha)$ on $[a,b]$, $\alpha \in C([a,b])$ $$ \beta(x)=\int_a^xg(z)d\alpha(z) \text{ on [a,b]} $$ Why is the additional assumption $f$ continuous and what ...
0
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0answers
22 views

If $(X,d_1)$ and $(X,d_2)$ two connected metric spaces if only if $X\times Y$ is connected metric space

$(X,d_1)$ and $(X,d_2)$ are two connected metric spaces if and only if $X\times Y$ is a connected metric space with metric $$ D((x_1,y_2), (x_2,y_2)) = \max(d_1(x_1,x_2),d_2(y_1,y_2)).$$ I know that ...
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0answers
13 views

Dense convex set in $*$-weak topology

Let $X$ be a Hausdorff topological vector space over $\mathbb{K}$. Suppose $W$ is a convex subset of its topological dual $X'$. How to prove that if for any $x\in X\setminus\{0\}$ set $\{f(x):f\in ...
0
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1answer
15 views

Proving that a set infers a norm given certain conditions

Let $\bar{B}$ be a set in a vector space $E$. Then $\bar{B}$ is the closed unitary ball for a norm iff $\mathbb{N} . \bar{B} = E$ $\lambda x + (1-\lambda)y \in \bar{B}$ for all $x \, ,y \, \in ...
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0answers
15 views

Show that the function is identically Zero in certain subset

We are given a open ball D (radius = 1) in $\mathbb R^2$. and let $\{x_n\}$ be the dense sequence in the set D. Around each point $x_n$ we make a hole of radius $r_n$. The sequence $r_n$ satisfy the ...
0
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0answers
13 views

Determining the unit normal field of a paraboloid $P$, and integrating a vector field over $P$

Let $M \subseteq \mathbb{R}^n$ be a $n-1$-dimensional manifold, and $N_x M$ the normal vector space of $M$ at a point $x \mathbb{R}^n$, that is, the (1-dimensional) space of vectors that are ...
1
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2answers
47 views

homogeneous differential equations $y' = f(y/x)$

There is a weird Theorem that comes about when considering whether a function is homogeneous (in the sense of the title definition). I was unable to prove it, or to find a proof to it. Can any one ...
0
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1answer
25 views

prove $\sup X \le \sup Y$ if $X$ is a subset of $Y$ both sets are nonempty and $Y$ is bounded above

Working through foundations of mathematical anlysis by johnsonbaugh per suggestion and wondering if the following proof works? (no solutions to book) Problem: Let X and Y be nonempty subsets of real ...
0
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1answer
14 views

Continuous map from $L^r(\Omega)$ to $L^s(\Omega)$.

The following theorem appears in the appendix of P.H. Rabinowitz monograph on Critical Point Theory: Let $\Omega \subset \mathbb R^n$ be bounded. Let $g$ be such that (i) $g \in C(\overline{\Omega} ...
0
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0answers
16 views

Runge's Theorem Application

Below is a question out of Gamelin's Complex Analysis which I cannot quite figure out. Any tips would help appreciated! "Let $(z_j)$ be a sequence of distinct points in a domain $D$ that accumulates ...
0
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1answer
40 views

About the proof of the four colour theorem

The proof of the classification of finite simple groups is thousands of pages which does not seem to be a human readable proof because its too long (more than 10000 pages) to be read by one person ...
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2answers
50 views

Proof by induction: $(a+b)^n=a^n+na^{n-1}b+\frac{n(n-1)}{2!}a^{n-2}b^2+…+nab^{n-1}+b^n$

I'm studying in preparation for a Mathematical Analysis I examination and I'm solving past exam exercises. If it's any indicator of difficulty, the exercise is Exercise 1 of 4, part $a$ and graded for ...
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2answers
41 views

ELI5: Riemann-integrable vs Lebesgue-integrable

I am wondering what the difference between riemann-integrable and lebesgue-integrable means. Does it have anything to do with the absolute value of the integrand, something like ...
0
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0answers
45 views

Bound the first derivative of the following function: $f(x)=g(x)+h(x)$

Consider a decreasing function $f(x)$, resulting from the sum of two other decreasing functions: $f(x)=g(x)+h(x)$. All these 3 functions are positive. In addition, we have $g(0)=h(0)=1$. Further, we ...
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0answers
14 views

Volume Problem in Munkres' analysis on manifolds

I am having trouble with problem (a) of this question. I figured that the volume of $\triangle_1(R)$ is $|(\alpha(a+h, b)-\alpha(a, b))\times (\alpha(a+h, b+k)-\alpha(a, b))|$ but don't know how I ...
3
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1answer
56 views

How to prove that for all $k\in\mathbb N$, $h(kx)=kh(x)$ and $h(x+y)\le h(x)+h(y)$?

Suppose $X$ is a commutative monoid and $f:X\to\mathbb R\cup\{\infty\}$ a function and $$g(x)=\inf\left\{\sum_{i=1}^nf(x_i)~\middle\vert~\sum_{i=1}^nx_i=x,n\in\mathbb N\right\}$$ ...
2
votes
2answers
83 views

Theorem 2.43 in Baby Rudin: How to understand the proof?

Here's Theorem 2.43 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition: Let $P$ be a non-empty perfect set in $\mathbb{R}^k$. Then $P$ is uncountable. Here's the ...
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0answers
33 views

Show that $\lim_{z\to z_0} cf(z)=ac$ (where $c$ is a complex number)

How do I prove this? Suppose that $a, b$ and $c$ belong to $\mathbb C$ and that $$\lim_{z\to z_0} f(z)=a$$ and $$\lim_{z\to z_0} g(z)=b.$$ a - $\lim_{z\to z_0} cf(z)=ac$ (where $c$ is a ...
2
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2answers
36 views

Any necessary and sufficient condition(s) for closure of an open ball to be the corresponding closed ball?

Let $(X,d)$ be a metric space, $a \in X$, and $\delta$ be a positive real number. Then the open ball $B(a;\delta)$ is defined as $$B(a;\delta) \colon= \left\{ \ x \in X \ \colon \ d(x,a) < \delta ...
3
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1answer
55 views

Prove if $\displaystyle \sum_{n=1}^ \infty a_n$ converges, {$b_n$} is bounded & monotone, then $\displaystyle \sum_{n=1}^ \infty a_nb_n$ converges.

Prove that if $\displaystyle \sum_{n=1}^ \infty a_n$ converges, and {$b_n$} is bounded and monotone, then $\displaystyle \sum_{n=1}^ \infty a_nb_n$ converges. No, $a_n, b_n$ are not necessarily ...
0
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1answer
39 views

How to bound this complex number from below?

I am doing an $\epsilon-\delta$ proof ($z \rightarrow i, f(z) \rightarrow \infty$) and currently have the absolute value $$|f(z)|=\left|\frac{z-1}{z^2+1}\right|$$ and I wish to make a statement about ...
2
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5answers
33 views

Prove a function is in Big-Oh and not in Big-Omega

We are told to use the definitions of Big-Oh and Big-Omega to prove that a given function is in $O(f(n))$ or $\Omega(f(n))$. It requires being able to use $c$ and $n_0$. Use the definitions to show ...
3
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2answers
32 views

Composition involving bounded linear operators

I recently come across the following statement mentioned in a proof: Let $X,Y$ be normed linear spaces and $T:X \rightarrow Y$ be a linear operator. if for every bounded linear functional $U: Y ...
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0answers
26 views

Convex function from condition

Can we deduce that $F$ is a convex function (i.e $F''(t)>0, \forall t>0$) from the following conditions: $F(t)=\int_{0}^t f(\xi) d\xi$, $0\leq \theta F(t)< t f(t), \forall t>0$ The ...
2
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3answers
59 views

Why doesn't the limit $\lim_{(x,y) \rightarrow (0,0)} \frac{ e^{x+y} - x - y}{\sqrt{x^2 + y^2}}$ exist?

Why is this limit non-existant? $\lim_{(x,y) \rightarrow (0,0)} \frac{ e^{x+y} - x - y}{\sqrt{x^2 + y^2}}$ I can't seem to find $2$ different paths that would show it is non-existant.
0
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1answer
12 views

infimum of operator norms of iterations of linear operators

I am currently reading a proof in which a fact is used without proof: For a Banach space $X$ and a bounded linear operator $T: X \to X$, $$ \lim_{n \to \infty} \| T^n \|^{\frac{1}{n}} = \inf_{n ...
2
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1answer
38 views

How would you approach the limit $\lim_{z \rightarrow 0} \frac{ \sin ||z||_p}{||z||_p}$? [on hold]

How would I solve the limit $\lim_{z \rightarrow 0} \frac{ \sin ||z||_p}{||z||_p}$ ? Note that $p \in [1,\infty]$.
3
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3answers
59 views

Is there anything wrong with this definition of discontinuity?

Is there anything wrong with this definition of discontinuity for a function y = f(x)? $\forall \delta>0\, \exists \varepsilon>0$ such that $\vert x-c\vert < \delta$, but $\vert f(x) - ...
0
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1answer
40 views

How to prove $\displaystyle\bigcup^\infty_{k=1}(\bigcap^\infty_{n=1}A_{k,n})\subset\bigcap^\infty_{n=1}(\bigcup^\infty_{k=1}A_{k,n})$

Want to show $$\displaystyle\bigcup^\infty_{k=1}\left(\bigcap^\infty_{n=1}A_{k,n}\right)\subset\bigcap^\infty_{n=1}\left(\bigcup^\infty_{k=1}A_{k,n}\right)$$ Note the bottoms are $k=1,n=1$ and ...
4
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1answer
57 views
+250

Construction of a continuous function which maps some point in the interior of an open set to the boundary of the Range

I was studying the Inverse function theorem when I came across the following problems : (Let the closed set $V$ i.e the range have non-empty interior) Does there exist a continuous onto ...
5
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7answers
141 views

Convergence of the recursive sequence $a_{n+1}=\sqrt{2-a_n}$

I'm attempting to prove the convergence (or divergence, though I strongly suspect it converges) of a sequence defined as $a_{n+1}=\sqrt{2-a_n}$ with $a_1=\sqrt{2}$. I cannot use the monotonic ...
1
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1answer
22 views

intersection of intervals from 1 to infinity

This is a continuation of my previous quesion concerned with finding $ \cup^\infty_{k=1} S_k $ for $ S_k = (1 − 1/k, 2 + 1/k], k \in N $. And the answer intuitively makes sense, but what about for ...
0
votes
2answers
24 views

proving union of interval equals interval

Consider the interval $ S_k = (1 − 1/k, 2 + 1/k] $ and I want to find $ \cup^\infty_{k=1} S_k $ with proof How do I go about this? What I was thinking... as $ k \rightarrow \infty $ $ 1/k ...
2
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2answers
23 views

Supremum not in interior of set $A$

If $ A \neq \emptyset $ and is bounded above and $ s = \sup A $ then $ s \in \bar A $ and $ s \notin \operatorname{int}A$ I've got the first part, showing that $s$ is a limit point of $A$. Thus, $ s ...
0
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1answer
23 views

Why is it that for any rational numbers $a < b$, the interval $[a, b]$ in $\mathbb{Q}$ is not compact with respect to this metric?

Suppose $q$ is any nonzero rational number and $p$ is a fixed prime. If $q = p^k\frac{n}{m}$ for integers $n$ and $m$, neither of which has $p$ as a factor, then we define $|q|_p := p^{−k}$. We can ...
0
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1answer
22 views

How can I show that the sequence $a_n := p^n$ is a convergent sequence in this metric and find its limit?

Suppose $q$ is any nonzero rational number and $p$ is a fixed prime. If $q = p^k\frac{n}{m}$ for integers $n$ and $m$, neither of which has $p$ as a factor, then we define $|q|_p := p^{−k}$. We can ...
1
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3answers
30 views

Why is $\sup_{x∈[0,1]} {|p'(x)|} ≤ A_d\sup_{x∈[0,1]}{|p(x)|}$ for all polynomials $p$ of degree at most $d$?

How can one prove that for any positive integer $d$, there is a constant $A_d < 0$ such that $$ \sup_{x∈[0,1]} {\lvert\, p'(x)\rvert} ≤ A_d\sup_{x∈[0,1]}{\lvert\, p(x)\rvert}, $$ for all ...
1
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0answers
18 views

Why are $F(p) := \sup_{x∈[0,1]}{|p'(x)|}$ and $G(p) := \sup_{x∈[0,1]}{|p(x)|}$ both continuous functions of the polynomial $p$?

Why are $F(p) := \sup_{x∈[0,1]}{|p'(x)|}$ and $G(p) := \sup_{x∈[0,1]}{|p(x)|}$ both continuous functions of the polynomial $p$, which is finite and of degree at most $d$ ? Continuity of a function ...
-1
votes
2answers
18 views

Why do the coefficients of all polynomials of degree at most $d$ as coordinates of a vector in $\mathbb{R}^{d+1}$ lie in ${R}^{d+1}$'s unit sphere?

Consider the coefficients of all polynomials of degree at most $d$ as coordinates of a vector in $\mathbb{R}^{d+1}$. Why would it suffice to suffices to assume that this vector lies in the unit ...
0
votes
1answer
25 views

For a sequence, why must $\lim _{n→∞} {||x_n||} = ∞$, $\lim _{n→∞} {||x_n||} = 0$, or there exists a convergent subsequence with a nonzero limit?

Suppose I've got a sequence of vectors $\{x_n\}_{n∈N}$ in $\mathbb{R}^k$. Why is it that exactly one of the following three facts must hold: $\lim _{n→∞} {||x_n||} = ∞$, $\lim _{n→∞} {x_n} = 0$, or ...
-1
votes
0answers
18 views

Proof of Well-ordering principle [on hold]

In analysis class we are building natural numbers from the real numbers The teacher gave us a Theorem Theorem(Well-ordering principle) Let $G \subseteq \Bbb{N}$ non-empty then $\exists m_0$$\in G: ...
0
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0answers
25 views

Prove or Disprove If $T : R^n \rightarrow\ell_2$ is linear, then $T(A)$ is totally bounded in $\ell_2$ when $A ⊆ R^n$ is bounded

Prove or Disprove, If $ T: \mathbb R^n \rightarrow \ell_2 $ is linear, then It preserves total boundedness $ T(A) $ is totally bounded in $\ell_2$ when $A \subseteq\mathbb R^n$ is bounded. I think ...
2
votes
1answer
16 views

Continuity on the parameters of the intermediate value theorem

Let $X$ be a compact metric space (feel free to impose more conditions as long as they're also satisfied by spheres) and $F : X \times [0, 1] \to \mathbb{R}$ a continuous function such that $F(x, 0) ...