Mathematical Analysis generally, including Real Analysis, Harmonic Analysis, Complex Variable Theory, the Calculus of Variations, Measure Theory, and Non-Standard Analysis.
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0answers
32 views
Complete normed vector space
I came across this question and was not sure how to proceed. Is the normed vector space of all polynomial functions on [0,1] complete with respect to the infinity norm? Thank you
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1answer
40 views
How do I prove that an uncountable subset $S$ of $\mathbb{R}$ has the in-between property?
The in-between property is that between any two distinct reals in the set, there is another real number. Also, $S$ has no discontinuities. It's not an interval such as $[0, 1] \cup [2, 3]$, for ...
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0answers
61 views
Geometrical Inequality
Let $ABCD$ be a quadrilateral on the unit circle, and the diagonals
$AC$ and $BD$ intersects at $E$. If the shortest height of the
triangle $ACD$ equals the radius of the incircle of the triangle ...
0
votes
2answers
18 views
Affine maps problems
How to find out a particular affine map when some points are given, say if it takes (0,0) to (1,1), (1,0) to (3,2) and (0,1) to (2,4)?
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votes
4answers
103 views
How to find the limit of $\dfrac{\ln(\ln(\frac{n}{n-1}))}{\ln(n)}$?
How do you find
$$\lim_{n \to\infty} \dfrac{\ln(\ln(\frac{n}{n-1}))}{\ln(n)}$$
I know it's $-1$, but I had to plot it.
4
votes
1answer
47 views
Suppose that the function f(x)
Suppose that a function $f(x)$ defined on $[0,1]$ satisfies
$f(1/n)\to 0$ as $n\to\infty$. Can we say that $f(x)\to 0$ as $x\to 0^+$ if $f$ is continuous on $[0,1]$ ? and again
is it true $f(x)\to ...
3
votes
1answer
64 views
Proof to show function f satisfies Lipschitz condition when derivatives f' exist and are continuous
The question is as follows:
Given a function f, 2 known information:
(1) $f'(x)$ exist
(2) $f'(x)$ are continuous
Goal: function f satisfies Lipschitz condition on any ...
4
votes
1answer
37 views
Can anyone show or clarify
Can any anyone clarify or prove that if the derivative of a function $f$ is strictly positive then the function $f$ is strictly monotone increasing. I am really sure that the converse is not true as ...
5
votes
2answers
50 views
Proof using Rolle's Theorem to show there is c such that f$^4$(c) = 0, for a < c < b
The question is as follows:
Give 3 information:
(1) f is a polynomial (thus I claim f is continuous at every point)
(2) $f(a) = f'(a) = f''(a) = f'''(a) = 0$
(3) $f(b) = 0$
...
3
votes
1answer
33 views
Is a continuous function in two variables necessarily equicontinuous in one variable?
Suppose $K \in \mathcal{C}\left(\left[0, 1\right]\times\left[0, 1\right]\right)$. Then, is it necessarily the case that the set of functions $\left\{g_y(x):g_y(x) = K(x,y), \forall y \in ...
2
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1answer
67 views
assume $f:[a,b] \to \mathbb R$ such that $f '(a)=f '(b)$ how prove $\exists t\in(a,b) : f(t)-f(a)=f '(t)(t-a)$?
let $f:[a,b] \to \mathbb R$ such that $f '(a)=f '(b)$ how prove $$\exists t\in(a,b) : f(t)-f(a)=f '(t)(t-a)$$
Thanks in advance
3
votes
3answers
103 views
Prove that there is at least one real solution to the equation…
$x^{17}+\frac{243}{1+x^4}=120$
Can anyone show me how to approach this problem..? Any help would be great, thanks.
1
vote
1answer
76 views
Convexity of $x^2f(x)$
Given a function $f$ which is decreasing and convex on $(0,\infty)$, is it possible to find a simple condition on $f$ such that
\begin{equation}
2f(x) + 4xf^\prime(x) + x^2f^{\prime\prime}(x) \geq 0.
...
0
votes
1answer
25 views
Analysis - finding local extrema?
I must find and identify (max or min) the local extrema of $f(x) = x^2 e^{-x}$
This is a simple problem if it was in a calculus exam - but it's not. I'm not sure how to structure the solution for an ...
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votes
2answers
34 views
Question regarding infinite sets of a metric space.
Let $A_1, A_2, A_3, ...$ be subsets of a metric space $X$.
(a)If $B_n = \cup_{i=1}^n A_i$, prove that closure $\overline {B_n} = \cup_{i=1}^n \overline {A_i}$.
(b)If $B = \cup_{i=1}^{\infty} A_i$, ...
6
votes
6answers
209 views
To construct a set with a limit point.
I learned how to construct a Cantor Set, and I am asked to do the following.
"Construct a bounded set with exactly 3 limit points."
Since the Cantor set contains infinitely many points, I don't ...
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votes
2answers
42 views
Problems regarding countable sets.
I am required to prove that the set of algebraic numbers is countable.
My understanding of an algebraic number is the following.
(1) A solution $z$ to the equation ...
1
vote
1answer
44 views
Proof on showing a uniformly continuous function has limit at every cluster point of the domain
The question is as follows:
Given:
(a) f is uniformly continuous on a subset D of $\mathbb R^n$
and (b) $x_0$ is a cluster point of D
Show: The limit of f(x), as x approaches ...
0
votes
1answer
49 views
Roots of a polynomial satisfying $f(x^{2}+1) = f(x) \cdot g(x)$
Let $f(x), g(x)$ be $2$ real polynomials of degrees ($m\ge 2$) and $(n\ge 1 )$ respectively satisfying $$f(x^{2}+1) = f(x) \cdot g(x)$$
for every $x \in \mathbb{R}$. Then which of the below options ...
1
vote
1answer
146 views
Show there exists a sequence of positive real numbers s.t. …
Let $f_n$ be a sequence of measurable functions on $[0,1]$ with $|f_n(x)|\lt\infty$ a.e. Show there exists a sequence $c_n$ of positive real numbers s.t. $f_n(x)/c_n\to0$ for almost every $x$ in ...
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votes
1answer
34 views
About differentiability and partial differentials of function.
Problem Statement:
Given:$$f: \mathbb {R^2} \rightarrow \mathbb {R},(x,y)\rightarrow \begin{cases} 0 & (x,y)=(0,0) \\ \frac{x^3}{x^2+y^2} & (x,y)\neq (0,0)\end{cases} $$
Need to show that it ...
1
vote
1answer
48 views
Show that this is a diffeomorphism
I have a function $F:(0,2\pi) \times \mathbb{R}_{>0} \rightarrow \mathbb{R}_{>0}^2$
with $(\phi,r)\mapsto(r(\phi-\sin(\phi)),r(1-cos(\phi)))$ and want to show that this is a smooth(meaning ...
0
votes
1answer
33 views
Nondegenerate critical point
I don't understand this part from the book of Zeidler , can someone help me to understand it ?
Please
Thank you
2
votes
1answer
32 views
proving differentiability for a function in a point
Given a differentiable function $f:D\backslash\{a\}\rightarrow\mathbb R$ and $\lim_{x\rightarrow a}f'(x)=c$ and $f$ is continous in $a$, I want to prove that $f$ is differentiable in $a$ and ...
5
votes
1answer
62 views
Class $C^{- \infty }$ functions?
If my understanding is correct, a class $C^{-1}$ function (in terms of smoothness, of course) can be thought of as a function which integrates to a class $C^{0}$ function. And when we differentiate ...
3
votes
0answers
28 views
Some questions regarding Ramanujan summation — Part I
The Ramanujan Summation method, is a method through which divergent series can be summed to convergent values.
I have several questions regarding this summation method. For more info about the words ...
1
vote
1answer
23 views
Are Trigonometric Functions Dense in $C^k(S^1)?$
Consider the functions $\{e^{2\pi i nx}\}_{n \in \mathbb{Z}}$ defined on the interval $[0,1].$ These are all smooth periodic functions (so functions on $S^1)$ and by the Stone-Weierstrass theorem ...
6
votes
1answer
77 views
Minimal definition of the derivative
The definition of the Fréchet derivative according to Wikipedia is:
Let $V$ and $W$ be Banach spaces, and $U\subset V$ be an open subset of $V$. A function $f : U \to W$ is called Fréchet ...
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votes
3answers
69 views
Function Continuity on an Interval.
I must show that $f(x)=p{\sqrt{x}}$ , $p>0$ is continuous on the interval [0,1).
I'm not sure how I show that a function is continuous on an interval, as opposed to at a particular point.
2
votes
3answers
58 views
Basic topology question regarding the complex plane.
Prove that the Complex plane is closed, open and perfect.
My intuition is destroyed by the fact that a set can be open and closed at the same time.
The following is my understanding.
open: If all ...
3
votes
2answers
54 views
A minor question about the Cantor Set
I'm self teaching analysis and the second chapter is about some basic topology.
According to the book "Principles of Mathematical Analysis (3rd)" from Walter Rudin,
the Cantor Set is constructed as ...
1
vote
0answers
42 views
Deformation retract
How to prove that $r_t$ is a deformation retract
$M^a=\lbrace q\in M ; f(q)\leq a\rbrace$
We have the definition :
$r_t$ is a difformation retract if:
$r_t$ is a continius ,onto application ...
1
vote
1answer
53 views
Computing an explicit solution to an integral equation via the Neumann Series.
I am hoping that someone can provide guidance for solving the integral equation
$$u=f+\lambda Au$$
where $1/\lambda\notin\sigma(A)$, $f\in L^2[0,2\pi]$, and $A:L^2[0,2\pi]\to L^2[0,2\pi]$ is defined ...
1
vote
2answers
29 views
Minima for a Sum
Let $A=\{|a_i|:a_i\in\mathbb{Z}\land1\leq i\leq n\}$ and $n\geq 1$
Let $b_i=\frac{\max A}{|a_i|}.$
How can one prove that the minimum possible value for$\sum\limits_{i=1}^n b_i$ is $n$?
3
votes
1answer
65 views
A combinatorial identity with Pochhammer's symbol
Let $m,k$ be an positive integers with $k\le m$. I am trying to prove $$\sum_{j=0}^k{\frac{1}{2}\choose k-j}\frac{2^{2j}(m+j)!}{(m-j)!(2j)!}=\frac{P(n,k)}{(2k)!}$$
where $n=2m+1$ and ...
2
votes
3answers
72 views
Prove uniform convergence of $x^{\frac{1}{n}}+(1-x)^{n}$
Is it true or not that the this succession converges uniformly on $(C[0, 1],\Vert . \Vert_{\infty})$: $$f_{n}=x^{\frac{1}{n}}+ (1-x)^{n}$$I have found an elementary solution, but I would like to ...
3
votes
4answers
143 views
Finding a continuous function $f: \mathbb{R} \to \mathbb{R}$ such that $f(\mathbb{R})$ is neither open nor closed
Find a bounded, continuous function $f: \mathbb{R} \to \mathbb{R}$ such that $f(\mathbb{R})$ is neither open nor closed?
2
votes
1answer
58 views
A combinatorial identity related to Chebyshev differential equation
Let $m,k$ be an positive integers with $k\le m$. Does anyone have a proof that $$\sum_{j=k}^m {2m+1\choose 2j+1}{j\choose k}=\frac{2^{2(m-k)}(2m-k)!}{(2m-2k)!k!}?$$
This is related to Chebyshev ...
1
vote
1answer
55 views
Prove that $X_{v+w} \subset X_v+X_w$
Let $B \subset \mathbb{R}^d$ a set convex and simetric, ($B=-B$). Prove that $X_{v+w} \subset X_v+X_w$, where
$$X_v= \{ \alpha >0 \ ; \ \frac{1}{\alpha}v \in B \}$$
$$X_w=\{\varepsilon >0 \ ; \ ...
2
votes
1answer
95 views
Functional equation
Can someone help me please with this problem?
If the function
$f:\mathbb{R}^+\rightarrow\mathbb{R}$ satisfies the equation
$f\Big(\frac{x+y}{2}\Big)+f\Big(\frac{2xy}{x+y}\Big)=
f(x)+f(y)$,
then it ...
2
votes
1answer
99 views
The range of the derivative of a differentiable function
I read somewhere that, given a function $f$ differentiable on $[a,b]$, the range of $f'$ can be
(1) a closed interval or
(2) an open interval or
(3) a half-open interval or
(4) an unbounded interval
...
1
vote
2answers
94 views
Integral with hyperbolic functions
I need to compute:
$$ \int_{x^2+y^2=1} \frac{\sinh(x)dy- \sin(y)dx}{\cosh(x)-\cos(y)}$$ where the circle $x^2 + y^2 = 1$ is oriented anticlockwise. So, can somebody show me how? I found the ...
0
votes
1answer
18 views
Upper bound for $u_n(z) = n^{-2}\sec (\pi z / 2n)$
I've been working through past exam papers for an Analysis exam that's coming up. This question has had me tearing my hair out, and I would appreciate some help:
Use the Weierstrass M-test to prove ...
1
vote
0answers
36 views
Compactness in $L^p$
I am studying this article:
http://arxiv.org/pdf/0906.4883.pdf
There is a little part that I do not understand, in the proof of theorem 5, page 4.
Let P be the projection map of $L^p(\mathbb{R}^n)$ ...
1
vote
1answer
53 views
Highest derivatives of implicit function
I am learning to use the implicit function theorem (IFT) and met recently the following problem:
Let $F(x,y)=x+y+x^5-y^5$. The given equation defines a smooth function $\phi:U\rightarrow \mathbb{R}$ ...
1
vote
0answers
37 views
Series expansion of $\sin(n\arccos(x))$
Let $n=2m+1$ be an odd positive integer. Is there a clever way to prove that the Maclaurin series of $\sin(n\arccos(x))$ is equal to $$(-1)^m\left(1+\sum_{k=1}^\infty ...
1
vote
1answer
44 views
Prove for continuous f and g, f(x)<g(x) there exists k such that f(x)+k<g(x)
Suppose that $f$ and $g$ are continuous on $[a,b]$ and for each $x$, it holds that $f(x)<g(x)$. Prove that there exists $\alpha>0$ such that for each $x$, it holds that $f(x) + \alpha <g(x)$ ...
2
votes
1answer
37 views
Unique solution differential equation proof
Prove that there is a $\delta>0$ such that there is a unique solution of the differential equation $y'(t)=\sin(y(t))$ with $y(0)=1$ on the interval $[-\delta, \delta]$. How large can you choose ...
3
votes
1answer
48 views
$f(x) = e^{-x^2}$ series representation
Let $f(x) = e^{-x^2}$, defined for $x \in \mathbb{R}$. Find a series representation of a function $F:\mathbb{R} \to \mathbb{R}$ such that $F(0)=0$ and $F'(x)=f(x)$ for each $x$.
I know that the ...
2
votes
1answer
42 views
Many partitions of unity on sufficiently “nice”; what does this mean?
In a class I am taking, we are told that are manifolds have "many" partitions of unity (we assume paracompact, Hausdorff, second-countable). However, the course content is not related to this subject. ...
