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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11
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1answer
2k views

Why is the graph of a continuous function to a Hausdorff space closed?

Say I have two topological spaces given by $(X,\mathscr{T}_X)$ and $(Y,\mathscr{T}_Y)$ where $Y$ is Hausdorff. In addition say I have a function $f:X\rightarrow Y$, and let it be continuous. I want to ...
1
vote
2answers
390 views

product rule for matrix functions?

Given a real rectangular matrix $X$, and two scalar-valued matrix functions, $f(X)$ and $g(X)$, does the product rule for differentiation of a product of scalar valued functions, hold when ...
1
vote
3answers
86 views

A condition that balls have finite measure

Let $(X,d)$ be a metric space and let $\mu$ be a positive measure on $X$. I want to require that $(X,d)$ and $\mu$ have either of the following properties: $\forall y \in X$, $\forall r \geq 0$, ...
5
votes
2answers
110 views

Find all continuous functions (did I do this correctly?)

Find all the continuous functions $f:\mathbb{R}\to\mathbb{R}$ satisfying: $f(x+y)=f(x)+f(y)+f(x)f(y)$ for all $x,y\in\mathbb{R}$ Solution attempt: $$\begin{align*} f(x+y) + 1 &= f(x) + ...
1
vote
2answers
147 views

Necessary and Sufficient Condition for two metrics to have same open sets.

There are couple of independent conditions like one being scalar multiple of another, or if $$d_p(x,y)=(x^p+y^p)^{1/p}$$ then all $d_ps$ and $d_qs.$ which guarantee that open sets are same under these ...
3
votes
1answer
35 views

Calculating $d\omega$ for $\omega\in\Omega^{k}M$ explicitly for $k=2$

I am trying to explicitly calculate the exterior derivative $d\omega$ for $\omega\in\Omega^{2}M$ for a differentiable oriented manifold $M$. I know that we can express a differential $k$-form ...
5
votes
3answers
103 views

Intermediate value-like theorem for $\mathbb{C}$?

Is there an intermediate value like theorem for $\mathbb{C}$? I know $\mathbb {C}$ isn't ordered, but if we have a function $f:\mathbb{C}\to\mathbb{C}$ that's continuous, what can we conclude about ...
2
votes
2answers
179 views

Part of proof 11.10 in Rudin's Principles of Mathematical Analysis

There is a part of proof 11.10 that I don't get in Rudin's Principles of Mathematical Analysis (3rd edition). The whole theorem is the following two statements: $\mathcal{M}\left(\mu\right)$ is a ...
5
votes
3answers
80 views

Showing a function is not one-to-one near the origin

Let $$f(x)=\begin{cases} x+2x^2\sin\left(\frac{1}{x}\right) \text{ if } x \neq 0 \\ 0 \text{ if } x=0 \end{cases}$$ I'm trying to show this is not one-to-one near $0$. I was given a hint to consider ...
1
vote
0answers
40 views

Find the maximal and minimal values of the following function

Let $1<\beta_1\,,\beta_2<2$, define $f(\beta_1,\beta_2)=\frac{(\beta_1+\beta_2)(\beta_1^2+\beta_2^2)+(\beta_1\beta_2)^3}{\beta_1^2+\beta_1\beta_2+\beta_2^2}$, find the range of ...
1
vote
1answer
293 views

topology in R infinity

What does the following sentence mean and why is that true: "The nonnegative orthant in $R^{\infty}$ has empty interior in product topology" Thank you!
1
vote
1answer
74 views

infinite sum limit how to find the following

Hi what is the limit of the following sum: $$\lim \limits_{n\rightarrow\infty}\frac{2}{n^2}\sum\limits_{j=0}^{n-1}\sum\limits_{k=j+1}^{n-1}\frac{k}{n}$$ Thanks a lot!
1
vote
3answers
361 views

How to show that an open map $f $ implies the surjectivity of $f'$ in a dense set

Let $f$ be a $C^1$ map from $U\to \mathbb{R}^m$, where $U$ is an open set in $\mathbb{R}^n$, $n\geq m$. Then we know that if $f'$ is surjective everywhere, then $f$ is open. My question is whether ...
2
votes
1answer
65 views

A closed form for a particular topology.

I am trying to find some sort of 'closed form' (if possible) of a particular topology generated by the sets: $({x\in\mathbb{R}\ \vert x\geq a}), a\in \mathbb {R}$. Thanks !
1
vote
2answers
377 views

Showing a contraction without a fixed point

Suppose $f: [1, \infty) \to [1, \infty]$ defined by $f(x) = x + \frac{1}{x}$ for all $x \geq 1$. I want to prove that: \begin{equation} |f(x)-f(y)| < |x-y| \end{equation} except when $x=y$, but ...
11
votes
1answer
411 views

Does a nonlinear additive function on R imply a Hamel basis of R?

A function is additive if $f(x+y) = f(x) + f(y)$. Intuitively, it might seem that an additive function from R to R must be linear, specifically of the form $f(x) = kx$. But assuming the axiom of ...
1
vote
1answer
82 views

sequences of functions $\cos(\frac{x}{n})$

I have the sequence $(f_n)$ of functions $f_n: [-1, 1] \rightarrow \mathbb{R}$, defined by $f_n(x) = \cos\left(\frac{x}{n}\right)$. I need to show that $(f_n)$ is Cauchy in the space $(C[-1, 1], d)$ ...
0
votes
0answers
89 views

Riesz Representation Theorem and Indicator Function

I've been dealing with the Riesz Representation Theorem for measures and it is obvious that having a measure $\mu$ I can get a continuous linear functional $\mu^*$ in $C(X)^*$ where $X$ is a compact ...
6
votes
2answers
324 views

Can one define the derivative of a function using tangent cones? Does such a notion already exist?

I'm interested in finding an analogue of a derivative that applies to functions which are defined more general subsets of $\mathbb{R}^n$ than open subsets. In particularly, I'm looking at functions ...
1
vote
1answer
238 views

Complex differentiability implies real differentiability

First, we think of $\mathbb{R}^2$ as related to the complex plane by the following: $(x,y) \leftrightarrow x+iy$. Show that if $f(x,y)=(u(x,y),v(x,y))$ is complex differentiable at $z_0$, then $f$ is ...
12
votes
8answers
22k views

What is a simple example of a limit in the real world?

This morning, I read Wikipedia's informal definition of a limit: Informally, a function f assigns an output $f(x)$ to every input $x$. The function has a limit $L$ at an input $p$ if $f(x)$ is ...
0
votes
2answers
127 views

Closed and open sets

By regarding the real numbers with their natural topology, my textbook says, that: $$ \left\{2 \pi n+\frac{1}{n}\;\bigg|\;n \in \mathbb{N} \right\}$$ is closed, which i understand, as every sequence ...
3
votes
1answer
99 views

An identity related to Legendre polynomials

Let $m$ be a positive integer. I believe the the following identity $$1+\sum_{k=1}^m (-1)^k\frac{P(k,m)}{(2k)!}=(-1)^m\frac{2^{2m}(m!)^2}{(2m)!}$$ where $P(k,m)=\prod_{i=0}^{k-1} (2m-2i)(2m+2i+1)$, ...
3
votes
0answers
287 views

Is the $ϵ,δ$ definition of a limit not well-defined?

I just watched this youtube video: http://www.youtube.com/watch?v=K4eAyn-oK4M He lays out his objections against the $ϵ,δ$ definition around 14 min. Here is the discription of the video: In ...
2
votes
1answer
328 views

A surjective map which is not a submersion

Is there an example of a smooth map between smooth manifolds which is surjective, but not a submersion? I feel there can't be one, but don't know of a proof. Nor do I know of a counter-example. ...
3
votes
2answers
42 views

Does the sequence converges?

I am trying to prove if the sequence $a_n=(\root n\of e-1)\cdot n$ is convergent. I know that the sequences $x_n=(1+1/n)^n$ and $y_n=(1+1/n)^{n+1}$ tends to the same limit which is $e$. Can anyone ...
0
votes
1answer
57 views

Differentiable Functions on Open Subsets of $\Bbb R^n$

Let $U\in\mathbb{R}^m$ open. Show that, in order $f:U\rightarrow \mathbb{R}^n$ is differentiable at $a\in U$ is necessary and sufficient that there is, for every $h\in\mathbb{R}^m$ with $a+h\in U$, ...
1
vote
0answers
149 views

Determining Complete Metric Spaces

I need to determine if $((0, 1), d)$ where $d(x, y) = |x^2 - y^2| \forall_{x,y}\in (0,1)$ My argument is as follows, take the sequence defined by $\dfrac{1}{n}$, then we know by $d(x, y)$ that ...
1
vote
2answers
167 views

Hausdorff space and Cantor's intersection theorem

$X$ is a Hausdorff space, $C_i$ is a non-empty closed subset of $X$ and $C_{k+1}\subseteq C_k$ , show that $\displaystyle \bigcap_{i\in \mathbb{N}} C_i$ is compact. I tried to prove by ...
0
votes
1answer
92 views

Correctness of Analysis argument with Cauchy sequences

Let $(x_n)$ and $(y_n)$ be Cauchy sequences in $(X, d)$. Show that $(x_n)$ and $(y_n)$ converge to the same limit iff $d(x_n, y_n) \rightarrow 0$ Proof $\rightarrow$ Suppose $(x_n) \to a$ and $(y_n) ...
2
votes
1answer
149 views

Cauchy Sequences and Analysis

Let $(x_n)$ and $(y_n)$ be Cauchy sequences in a metric space $(X, d)$. Show that the sequence $(d(x_n, y_n))$ is a cauchy sequence in $\mathbb{R}$. What is the significance of $\mathbb{R}$ in this ...
0
votes
1answer
376 views

Prove [$A \subset B$, and $B$ is bounded in $n$-space] $\implies$ [ diameter of $A \leq$ diameter of $B$]

The question is as follows: $A \subset B$, and $B$ is bounded in $n$-space Show: the diameters, $\operatorname{diam}(A) \leq \operatorname{diam}(B)$ I came up with the following 2 ...
2
votes
2answers
126 views

$x_n$ is the $n$'th positive solution to $x=\tan(x)$. Find $\lim_{n\to\infty}\left(x_n-x_{n-1}\right)$

$x_n$ is the $n$'th positive solution to $x=\tan(x)$. Find $\lim_{n\to\infty}\left(x_n-x_{n-1}\right)$.
0
votes
3answers
612 views

How to understand both closed and open set in topology?

We've defined the connectedness in topology in class in this way that a topological space is connected if the only both open and closed set is empty set or the whole set. Now I got the explanation ...
3
votes
1answer
924 views

Uniform continuous maps that converge uniformly to a function f. What can we say about f?

Here is the question: Let $X$ and $Y$ be metric spaces.Suppose a sequence of uniformly continuous maps $f_n : X \rightarrow Y$ converges uniformly to a map $f: X \rightarrow Y$.Does that imply that f ...
3
votes
2answers
151 views

Showing the function $f(x,y)$ is one by one

Yesterday, while teaching geometry, I was faced to a problem saying that the function below is an distance function: $$d(P,Q)=\Big|\ln\frac{\frac{x_1-c+r}{y_1}}{\frac{x_2-c+r}{y_2}}\Big|$$ where in ...
1
vote
1answer
127 views

Example of bijection from $\mathbb{Q} \to \mathbb{Q} \times \mathbb{Q}$

What would be an example of bijection between $\mathbb{Q}$ and $\mathbb{Q}\times \mathbb{Q}$. I can think of one: $x \mapsto (x,x+1)$ Does this work? I am not sure.<
2
votes
0answers
245 views

Help with Riemann Lebesgue Lemma Proof

I know there are many proofs out there, but I was asked to prove this in a particular way. I'm stuck on this step: Note: Let $[a.b]$ be compact interval in $\mathbb{R}$ and let $g$ be a continuous ...
1
vote
1answer
46 views

Product of Sets

I have a quick question regarding the interpretation of notation in topology. My notes state: Let $X:=\Pi_{\alpha \in A} X_\alpha$ where A is an indexed set. My interpretation is that $X=X_{\alpha ...
1
vote
1answer
74 views

Prove Cauchy sequence from

Can anyone prove from first principles, that $\{(2n-1)/n\}_{n}$ is a Cauchy sequence. Thank you all.
0
votes
2answers
752 views

Prove that $f$ is not Riemann integrable.

If a function $f:[-2,3]\to \mathbb{R}$ is defined by $f(x)=\begin{cases} 2|x|+1 \; ;\; \text{ if } x \in \Bbb Q \\ 0 \; ;\; \text{ if } x \notin \Bbb Q \end{cases}$ Prove that $f$ is not Riemann ...
1
vote
0answers
46 views

Sup and lim sup of a function defined by double series

It is unlikely that the following function has a closed form expression: $$f(t)=\sum_{n=0}^\infty\sum_{m=0}^\infty \frac{(-1)^{m+n}}{(2m+1)^3(2n+1)^3}\cos\left(\pi t \sqrt{(2m+1)^2+(2n+1)^2}\right)$$ ...
4
votes
1answer
311 views

Gradient of sum of products of matrix traces

For a matrix $X \in \Re_{n\times d}$ find the gradient of $\sum_{i,j}[\langle X_{i.},X_{j.} \rangle\operatorname{tr}(X^TA_{ij}X)]$ w.r.t $X$, where $A_{ij}=(e_i-e_j)(e_i-e_j)^T$ using the basis ...
1
vote
1answer
238 views

Recommended textbooks for analysis past 1st year grad material

My current background in analysis is approximately the material in Folland's Real Analysis. I've also read the Analysis text by Lieb and Loss and I also took a graduate level class on complex ...
1
vote
2answers
101 views

Diffeomorhism of manifold

This is one of the exam questions of the previous semester. I have studied these. But I didn't do this. Please show me how to solve this question. Thank you for help
1
vote
0answers
164 views

The Cantor Space and open, but not closed sets.

consider the space $\{0,1\}^{\mathbb{N}}$ of all infinite binary sequences, called the Cantor-Space. This space is metrizable with metric $$ d(u,v) = 2^{-(r-1)} \qquad \textrm{ where } r = ...
2
votes
0answers
84 views

Closed form formula for a double series related to wave equation

Does anyone have a closed form formula for the double series $$\sum_{n=0}^\infty\sum_{m=0}^\infty \frac{(-1)^{m+n}}{(2m+1)^3(2n+1)^3}\cos\left(\pi t \sqrt{(2m+1)^2+(2n+1)^2}\right)?$$ This is related ...
2
votes
1answer
367 views

L'Hôpital's rule for vector-valued functions

What are the traps, when using L'Hôpital's rule in multivariable calculus to determine the lim of an vector function? I heard there some more cases, where L'Hôpital's rule in multivariable calculus ...
1
vote
1answer
317 views

Analysis - Fourier Transforms - show that convolution of characteristic functions is continuous

I would appreciate any instruction on the following exercise from real and complex analysis: Suppose $A$ and $B$ are measurable subsets of $\Re^1$, having finite positive measure. Show that the ...
1
vote
1answer
34 views

Determine the limit of the expression $\frac {\sum_{k=1}^n \frac{1}{k} - (1-\frac{2}{n})\ln n}{\ln n}$ as $n \to \infty$

I would like to analyze the convergence of the expression $$a_n =\frac {\sum_{k=1}^n \frac{1}{k} - (1-\frac{2}{n})\ln n}{\ln n}$$ as $n \to \infty$. In particular, I'd like to know that, if the ...