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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1
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1answer
18 views

Limit vs interior definition of continuity

Suppose I have two topological spaces $X$ and $Y$ whose topologies are defined by interior operators $\text{int}_X$ and $\text{int}_Y$ respectively, as well as a function $f$ with domain $I$ (for ...
3
votes
2answers
38 views

An infinite dimensional normed linear space is the union of two disjoint convex sets

Let $X$ be an infinite dimensional normed linear space. I want to show that there exist two disjoint convex sets $C_1$ and $C_2$ such that $X=C_1\cup C_2$ and both $C_1$ and $C_2$ are dense in $X$. I ...
4
votes
2answers
38 views

dominated convergence for functions $\mathbb R^n\to\mathbb R^m$?

I do know the dominated convergence theorem for functions $f:\mathbb R^n\to\mathbb R$. Now let $U\subset\mathbb R^n$ and $f: U\to\mathbb R^m$. Is there any dominated convergence theorem for ...
1
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1answer
32 views

Lemma 3.5-3 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS: Is the set of non-zere Fourier co-efficients uncountable too?

Let $X$ be an inner product space, let $x \in X$ be non-zero, and let $M$ be an uncountable orthonormal subset of $X$. Then what can we say about the cardinality of the following set? $$ \{ \ v \in M ...
-3
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1answer
28 views

A problem of the limit of a serie [on hold]

I need a step-by-step solution to the following problem. Sorry, I have nothing done because I don't know how to approach the problem. Calculate the following limit: Thanks!
0
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1answer
10 views

Prove $f\in \mathscr{R}(\alpha)$ and $\int_a^b f\ d\alpha = f(s)$ with the following conditions.

If $a<s<b$, $f$ is bounded on $[a,b]$, $f$ is continuous at $s$, and $\alpha(x)=I(x-s)$, then prove that: $$f\in \mathscr{R}(\alpha)$$ and $$\int_a^b f\ d\alpha = f(s)$$ $I$ is a unit step ...
-5
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0answers
57 views

A not very easy problem… [on hold]

I leave a challenge, a derivative problem. Determine $\alpha \in \mathbb{R}$ such that $$ \arctan^2 (2x) - \alpha x \sin x = \mathcal{O}(x^4), \text{ as } x\to 0 $$
1
vote
0answers
11 views

Compactness of a convex collection

Given $\epsilon\in(0,1)$, suppose we have collection $\mathscr{C}(\epsilon)$ of multilinear polynomials in $\Bbb R[x_1,\dots,x_n]$ that on $\{0,1\}^n$ is in range $(-\epsilon,\epsilon)$ on $S_0$ while ...
0
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0answers
24 views

a question on Frechet derivative

Suppose the derivative of a functional is given by $\int_{\Omega}(\vec{v}.\nabla u)|\nabla u|^{p-2} \phi$ for $\phi\in W_0^{1,p}(\Omega)$, then what is the functional?.
0
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0answers
23 views

How is this convex set compact as well?

Given $\epsilon\in(0,1)$, supposing we have a collection $\mathscr{C}(\epsilon)$ of polynomials in $\Bbb R[x_1,\dots,x_n]$ that on $\{0,1\}^n$ takes on value $0$ on $S_0$ while being in range ...
3
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1answer
15 views

$\{\,z\in \mathbb C : \operatorname{Im}(z)>-1 , |z|<2\, \}$ onto upper half space $\{\,z\in \mathbb C : \operatorname{Im}(z)>0 \,\}$

I am search an one to one mapping that maps the domain $$\{\,z\in \mathbb C : \operatorname{Im}(z)>-1 , |z|<2\, \}$$ onto upper half space $$\{\,z\in \mathbb C : \operatorname{Im}(z)>0\, ...
0
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0answers
18 views

Proving Euler's theorem for homogeneous functions.

This problem is from Apostol's Mathematical Analysis. Let $f$ be defined on an open set $S$ in $R^n$. We say that $f$ is homogeneous of degree $p$ over $S$ if $f(\lambda x)=\lambda ^p f(x)$ for every ...
0
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1answer
16 views

Relationship between centralization and floating-point arithmetic

Suppose I have a problem of least squares where I have $n$ independents regressor variables $x_i$, and suppose I applied the process of centralization and scaling to this variable through ...
0
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0answers
19 views

How do I compute this metric projection?

I saw a result that says: Given a nonzero vector $a$ and the convex set $K:=\{y\in H: \langle a,y\rangle =\alpha, \alpha \in \mathbb{R}\}$ a hyperplane, then $$P_Kx=x-\frac{\langle ...
0
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2answers
27 views

Find the limit points and exterior points of the following

Let $X=\mathbb R$, with the usual metric on $\mathbb R$ and $A=((0,1)\cap \mathbb Q)\cup$ {$2,3$}. Find the limit points of $A$, exterior points of $A$, $A^o$, $\overline A$ and $\partial A$. Can ...
-1
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0answers
36 views

Show that subspace metric induces subspace topology [on hold]

Let $(X,d)$ be a metric space, let $\tau$ be the topology on $X$ induced by $d$ and $A \subset X$. Define $d_A: A \times A \to \mathbb R$ as $d_A(a,b)=d(a,b) \forall a,b \in A$ . Show that $d_A$ ...
1
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1answer
31 views

Show that $\tau_A$ is a topology on $A$

Let $(X,\tau)$ be a topological space and $A \subset X$. Let $\tau_A$={$A \cap U: U \in \tau$}. Show that $\tau_A$ is a topology on $A$. I know that I need to prove three properties to prove ...
2
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1answer
53 views

Please check my demonstration of de l'hopital's rule

I have demostrate the de l'hopital theorem but in some steps I'm not 100% sure; The theorem I demostrate is for: $\lim_{x\rightarrow a+} \frac{f'(x)}{g'(x)}=L \implies\lim_{x\rightarrow a+} ...
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votes
0answers
31 views

Show that $x_n \to x$ in $(X,d_f)$ iff $f(x_n) \to f(x)$

Let $(x_n)$ be a sequence in $X$ and $x \in X$. $(X,d)$ is a metric space and $f: X \times X\to $ is a bijection. Show that $x_n \to x$ in $(X,d_f)$ iff $f(x_n) \to f(x)$. ...
-1
votes
2answers
31 views

Show that $d_f$ is a metric on $X$ [on hold]

Let $(X,d)$ be a metric space, and let $f: X \to X$ be a bijection. Define $$d_f: X \times X \to \mathbb R $$ as $d_f(x,y)=d(f(x),f(y))$ $\forall x,y \in X$ Show that $d_f$ is ...
-6
votes
1answer
38 views

how to find all roots of a polynomial function? [on hold]

Let $f(x)$ be a polynomial of degree 5 and $f(|x|)$ has nine real roots, then how many real roots does $f(x)$ has? Five real roots Four positive roots One negative root Nothing can be said in ...
7
votes
2answers
46 views

limit of function $\sin(x \ln x)/x$ as $x\rightarrow 0$

I am trying to find $\lim \limits_{x \to 0} \frac{\sin(x \space \ln(x))}{x}$. I believe I have solved it using the squeeze theorem to determine: $\frac{-1}{x} \leq \frac{\sin(x \space ln(x))}{x} \leq ...
1
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1answer
21 views

using logical quantifiers to write that f approaching infinity DOES NOT tend to infinity

Is this the same as writing that the limit of f as f approaches $\infty$ is L? i.e.: $\forall \space \epsilon > 0 \space \exists \space c \space \forall \space x>c : |f(x) - L|< \epsilon$
0
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0answers
28 views

The closure of an open set in $\mathbb{R}^n$ is a manifold

I want to solve the following exercise from M. Spivak's Calculus on Manifolds (p. 114): (a) Let $A \subseteq \mathbb{R}^n$ be an open set such that boundary $A$ is an $(n-1)$-dimensional manifold. ...
0
votes
0answers
17 views

Condition for all derivatives to be L-Lipschitz

Let $f:\mathbb{R}\to\mathbb{R}$ be a function with infinitely many derivatives and let us use the notation $$ f^{(n)}(x)=\frac{\mathrm{d}^nf(x)}{\mathrm{d}x^n}. $$ Assume that $f^{(n)}$ is ...
1
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1answer
12 views

Why is a continuous function of a Riemann-Stieltjes Integrable function, also Riemann-Stieltjes Integrable?

Suppose $f\in \mathscr{R}(\alpha)$ on $[a,b]$, $m\leq f\leq M$, $\phi$ is continuous on $[m,M]$, and $h(x)=\phi(f(x))$ on $[a,b]$. Then prove that $h(x)\in\mathscr{R}(\alpha)$ on $[a,b]$.
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0answers
27 views

Proving that $\phi$ is orthogonal to the harmonic forms given $\int\phi \;d\mathrm{vol}$.

I want to prove that given a connected closed (compact without boundary) oriented Riemmanian manifold $(M,g)$, the condition $$\int_M \varphi \;\mathrm{d}\mathrm{vol}_g=0$$ implies that $\int \varphi ...
2
votes
1answer
13 views

Given a function $f$ defined in $R^2$. Let $F(r,\theta)=f(r\cos\theta,r\sin\theta).$ Verify a formula of the modulus of the gradient.

Given a function $f$ defined in $R^2$. Let $$F(r,\theta)=f(r\operatorname{cos}\theta,r\operatorname{sin}\theta).$$ Verify the formula $$|\nabla f(r\operatorname{cos}\theta, ...
1
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1answer
40 views

Show that S is closed but not compact

Show that $S$={$(x,y,z)\in \mathbb R^3: x^3+y^4-z^2=1$} is closed but not compact where $\mathbb R^3$ is the usual topology. Can anyone explain how to go about answering this? I have to show that ...
0
votes
1answer
14 views

Can we deduce if a set is measurable, given a measurable function and a measurable space?

Let $f(x):X\rightarrow Y $, where $X$ is a measurable space. Suppose that $f$ is measurable. Let $E$ be a subset of $X$. Now, suppose that $f(E)$ is closed or clopen. Can we deduce that $E$ is a ...
0
votes
1answer
12 views

Why does a Hermitian operator with singleton spectrum have to be scalar?

One proof of Schur's lemma proceeds by showing that a Hermitian intertwining operator of an irreducible representation (of a topological group on a Hilbert space) has a spectrum that contains only one ...
2
votes
2answers
56 views

A property for an ODE

$2\leq n\in\mathbb{N}$. I have no idea how to show that there is a unique solution $y\in C^1([0,T))$ of the ODE \begin{eqnarray} \begin{cases} y'(t)=(1+y(t)^2)\left(1-\dfrac{n-1}{t}y(t)\right)\ \ \ ...
0
votes
1answer
21 views

Are maximal intervals of open nonempty sets always equal?

Let $O\subset\mathbb{R}$ be an open nonempty interval. Define for every $x\in O$: $$a_x = \inf\{a\in\mathbb{R}\mid(a,x]\subset O\}$$ $$b_x = \sup\{b\in\mathbb{R}\mid[x,b)\subset O\}$$ $$I_x = (a_x, ...
0
votes
0answers
9 views

Consistent estimators/convergene in probability and slutsky

Let $m_n$ be a consistent estimator of $g(\vec\alpha)$ where $\vec\alpha = (\alpha_1,\cdots,\alpha_k)\in \mathbb{R}^k$ and $v_n$ be a consistent estimator of $f(\alpha_1,g(\vec\alpha))$. Suppose that ...
1
vote
1answer
37 views

Let $A,B \subset \mathbb{R}^n$ open sets, $A\subset B$. How can I set a function $f:\mathbb{R}^n\longrightarrow\mathbb{R}$, $C^{\infty}$

Let $A,B \subset \mathbb{R}^n$ open sets, $A\subset B$. How can I set a function $f:\mathbb{R}^n\longrightarrow\mathbb{R}$, $C^{\infty}$, such that: \begin{equation*} f(x)=0, \textrm{ if } x\in ...
3
votes
2answers
41 views

Find an example of a sequence not in $l^1$ satisfying certain boundedness conditions.

This question is about getting a concrete example for this question on bounded holomorphic functions posed by @user122916 (something that he really expected as explained in the comments). Give an ...
3
votes
1answer
21 views

Equivalency of the set of real numbers to the set of all continuous real functions?

I understand that the set of real numbers is equivalent to the set of real numbers in the interval $(0,1)$ and also equivalent to the set of all points in $\mathbb{R}^2$. I have seen a claim in a book ...
0
votes
1answer
21 views

Application of Residue theorem

Let f(z,w) be holomorphic in $\mathbb{C}^{n}$ and not identically zero on the w-axis. Let {$b_{j}$} be the set of singularities of f(z,w) in some disk of radius $|w| < r$. Why does the residue ...
3
votes
2answers
28 views

Equivalency of real numbers and points in the plane?

I understand that the set of real numbers is equivalent to the set of real numbers in the interval $(-1,1)$ by simply using $arctan$ function. However, I do not know how to find a one-to-one mapping ...
1
vote
1answer
15 views

define $f(\mathbf{x})=f_1(x_1)+\cdots +f_n(x_n)$. Show that $f$ has a differential at each point of an n-dimensional interval.

Given $n$ real-valued functions $f_1, \dots, f_n$, defined and having finite derivatives in the interval $(a,b)$. For each $\mathbf{x}$ in the $n$-dimensional interval $$S=\{(x_1,\dots ,x_n)\mid a\lt ...
4
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0answers
34 views

$\phi_{\epsilon} \ast \mu \rightarrow \mu$?

Let $\phi$ be a non-negative function on $\mathbb{R}$ with $\int_{\mathbb{R}} \phi = 1$. Define $\phi_{\epsilon}(x)=\epsilon^{-1}\phi(\epsilon^{-1}x)$ for $x \in \mathbb{R}, \epsilon > 0$. For $f ...
2
votes
2answers
50 views

Homeomorphism between the set of invertible matrices and itself

Consider the set of invertible $n \times n$-matrices $GL_n(\mathbb{R}) = \{A \in M_{n \times n}(R) \mid A\text{ is invertible}\}$. I now want to prove that the transformation $$f: A \mapsto A^{-1}$$ ...
-1
votes
1answer
59 views

How can I find out if a non-convergent series is “indeterminate” (that is, “oscillating”) or “divergent”?

Definitions: Given a sequence $\{a_n\}$, define $$s_n= \sum_{j=0}^n a_j.$$ The sequence $\{s_n\}$ is called the series of partial sums of $\{a_n\}$. A series is convergent if $\{s_n\}$ has ...
3
votes
1answer
33 views

Conformal maps onto open right half plane

On the Big Rudin there is the conformal map $$\varphi(z) = \frac {1+z}{1-z}$$ which sends $\{-1, 0, 1\}$ to $\{0, 1, \infty\}$. The book says: The segment $(-1, 1)$ maps onto the positive real ...
0
votes
1answer
24 views

Laplacian operator on $L^2(\Omega)$?

Let $\Omega\subseteq \mathbb R^n$ be an open subset and $\displaystyle \Delta:=-\sum_{j=1}^n D_j^2$ be the Laplacian operator. I have some questions concearning this operator: $(i)$ Does it map ...
5
votes
1answer
53 views

$f, \hat{f} \in L^{p}(\mathbb R) \cap C(\mathbb R) \implies |f(x)| \to 0$ as $|x| \to \infty$?

Suppose $f, \hat{f} \in L^{p}(\mathbb R) \cap C(\mathbb R)\cap L^{\infty}(\mathbb R), (1<p<\infty).$ My Question: Can we expect $\lim_{|x|\to \infty} |f(x)|=0$ ? (In other words, If $f$ and its ...
0
votes
2answers
33 views

Study: $\sum_{n=1}^\infty (\sin(\sin n))^n$, $\sum_{n=1}^\infty \frac{n \sin (x^n)}{n + x^{2n}}$, and $\sum_{n=1}^\infty \frac{ \sin (x^n)}{(1+x)^n} $

Let $x \in \mathbb{R}$. I have to study the convergence of the following three series: $$\sum_{n=1}^\infty (\sin(\sin n))^n$$ $$\sum_{n=1}^\infty \frac{n \sin (x^n)}{n + x^{2n}}$$ ...
0
votes
0answers
20 views

Cauchy Criteria for Series

We know that the Cauchy Criterion of a series is as follow: Theorem: A series $\sum\limits_{i=1}^{\infty}x_i$ converges iff for all $\epsilon>0$ there is an $N\in \mathbb{N}$ so that for all ...
0
votes
1answer
16 views

operator norm of a linear transformation, given by the transformation matrix

Consider $\mathbb{K}^n$, $\mathbb{K}^m$, both with the $||.||_1$-norm, where $\mathbb{K} = \mathbb{R}$ or $\mathbb{C}$. Let $||T|| = inf\{M ≥ 0: ||T(x)|| ≤ M ||x|| \space \forall x \in ...
0
votes
1answer
15 views

Additive function in $\mathbb{R}^n$ is continuous, and related subspaces compact

I want to show that the function: $A: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}^n, (x, y) \mapsto x + y$ is continuous. Also, why is it that if $K, L$ are compact subspaces of $\mathbb{R}^n$, ...