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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Local maximum and negative definiteness

I was looking at an old Berkeley preliminary exam problem (Fall,88) stated below; Prove that a real-valued $C^3$ function $f$ on $\mathbb{R}^2$ whose Laplacian $$\frac{\partial^2f}{\partial ...
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1answer
151 views

Extension of Hölder continuous function

Let $f:[0,c] \rightarrow \mathbb R$ be Hölder continuous with constant $M>0$ and power $p \in (0,1)$ and satisfies $f(0)=f(c)$. Let $g:[0,2c] \rightarrow \mathbb R$ be given by: $$ g(x)=f(x) \ \ ...
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2answers
132 views

Question about the Fréchet derivative

In a proof to show that $D(L\circ f)_a = L \circ Df_a$, where $f: U \subset E \rightarrow F$ and $L \in L_c(F,G)$. They take the following limit to show that the frechet derivative exists, ...
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1answer
185 views

Weak convergence discrete space

Let $X_n$, $n = 1, 2, 3, \ldots$, and $X$ are random variables with at most countably many integer values. Prove that that $X_n \to X$ weakly if and only if $\lim_{n \to \infty} P (X_n = j) = P(X = ...
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1answer
182 views

A question about Cauchy sequences

Let $\langle a_n\rangle$ , $\langle b_n\rangle$ , $\langle c_n\rangle$ be Cauchy sequences of rational numbers, and $\langle c_n\rangle$ is equivalent to $\langle a_nb_n\rangle$. Prove or disprove ...
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0answers
40 views

Density problem

$U$ is any open set of $\mathbb{R}$. We known that $C_0^\infty(U)$ is dense in $C^k(U)$. But what about, say $C_0^\infty((0,1))$ in $C^k([0,1])$?
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1answer
166 views

Constructing function as follows?

I'm having difficulties in following construction used in proof. If $u$ is continuous function on open set $\Omega \subset R^n$ and $ p \in R^n$ satisfies $$\displaystyle \limsup_{y\to x} ...
19
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2answers
414 views

Proving that $f(n)=n$ if $f(n+1)>f(f(n))$

How can we prove that if $f:\mathbb{N}\rightarrow\mathbb{N}$ is a function so that $f(n+1)>f(f(n))$ for all $n\in\mathbb{N}$ then $f(n)=n$ for all $n\in\mathbb{N}$?
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0answers
69 views

Lebesgue-Stieltjes integral as a generalized function

Given some convex function $f(x)$, $x >0$ we can define a distribution $F \in \mathcal{D}'(0,\infty)$ using Lebesgue-Stieltjes integral $$ \langle F, \varphi \rangle ...
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6answers
1k views

Is there a number that's right in the middle of this interval $(0, 1)$?

This might seem like a silly question, but is there a number that's right in the middle of this interval $(0, 1)$? And the half-open intervals: $(0, 1]$, $[0, 1)$? I know for a fully closed interval ...
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1answer
521 views

minimizer of a function

Consider $1<p<\infty $. Let's define the space: $L_{V}^{p}(-1,1)=\left \{ f:(-1,1)\rightarrow \mathbb{R}:\int_{-1}^{1}\left | f(x) \right |^{p}V(x)dx<\infty \right \}$ Consider the norm: ...
3
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1answer
222 views

Second derivative of convex function

Let $f(x)$, $x>0$ be a convex function. Then it's distributional second derivative is defined by the rule $$ \langle f''(x),\varphi(x)\rangle = \langle f(x), \varphi''(x)\rangle $$ for any ...
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1answer
232 views

continuous function and not differentiable [duplicate]

Possible Duplicate: Function example? Continuous everywhere, differentiable nowhere Is there any function that continuous in all places and not differentiable in all places? do you know a ...
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0answers
34 views

Can I say something about the $f_{xy}$ of such function?

I have a function $f(x,y)$ defining on $x>0,y>0$ satisfying that (1) $f>0$ (2) for $a>1, f(ax,ay)>af(x,y)$ (3) $f_x,f_y>0$ (4)$f_{xx},f_{yy}<0$ Can I say something about ...
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2answers
2k views

Direct sums and direct products

This question has been in my head for a while. And today it appears again when I am reading Arveson's book on $C^*$-algebras. He says Countable direct products of Polish spaces are Polish. ...
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1answer
554 views

Use Cauchy's Multiplication Theorem and the Binomial Theorem to prove $\exp(x+y)=\exp(x)\exp(y)$

I am to use Cauchy's Multiplication Theorem and the Binomial Theorem in order to prove $\exp(x+y)=\exp(x)\exp(y) $ but I have no idea where to begin. All I can think of doing is setting $\exp(x)$ ...
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1answer
49 views

Expressing a differential form in terms of a scalar function

We can express every k-form in the form $ \omega(x) = \sum_Id_Idx_I $ where $ I$ is k-tuple and $ d_I$ is just some scalar function of x. That's entirely understandable for me. But while reading ...
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2answers
796 views

Determine whether this series is absolutely convergent, conditionally convergent or divergent?

The series $ \sum_{n=1}^{\infty} \frac{(-1)^n n}{n^2 + 1} $; is it absolutely convergent, conditionally convergent or divergent? This question is meant to be worth quite a few marks so although I ...
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3answers
250 views

How to prove this partial derivative?

Consider $u:\mathbf{R}\times\omega\rightarrow\mathbf{R}$, where $\omega\subset\mathbf{R}^{n-1}$ is a bounded domain. For each $y\in\omega$ and each $\lambda>0$, consider ...
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2answers
75 views

Sufficient proof of differentiability?

If I am asked to prove that the function $x^3 \sin(\frac{1}{x})$ is differentiable for all $x \ne 0$, is it sufficient to say something like the following: We see that $x^3$, $\sin(\frac{1}{x})$ and ...
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3answers
263 views

Convergence of finite differences to zero and polynomials

Assume that $f:\mathbb R \rightarrow \mathbb R$ is continuous and $h\in \mathbb R$. Let $\Delta_h^n f(x)$ be a finite difference of $f$ of order $n$, i.e $$ \Delta_h^1 f(x)=f(x+h)-f(x), $$ $$ ...
5
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1answer
428 views

Line integral and integration of differential forms

The definition of integral of a $k$-form $\omega$ over a parametrized manifold $ Y_\alpha $ is $ \int_{Y_\alpha}\omega = \int_A\alpha^*\omega$ where $ \alpha\colon A \to R^n $. Let $ \gamma:(a, b) ...
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3answers
2k views

arithmetic mean of a sequence converges

We had a theorem that the means of a sequence also converges: Let $(a_n)_{n\in\mathbb N}$ be a convergent sequence. Then $\displaystyle \overline a_n=\sum_{k=1}^n \frac{a_k}n$ also converges. ...
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2answers
68 views

$(a_{2n})$ and $(a_{2n+1})$ converges then $(a_n)$ converges

Whe had the following theorem in class: If $(a_{2n})_{n\in\mathbb N}$ and $(a_{2n+1})_{n\in\mathbb N}$ are convergent sequences with the same limit $a$, then the sequence $(a_{n})_{n\in\mathbb N}$ ...
3
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2answers
255 views

Proving that $\sum_{|j| < n} (n-|j|) \exp(ij\lambda)= \frac{\sin^2(\frac 1 2 n\lambda)}{\sin^2(\frac 1 2 \lambda)}$

I want to show that $\sum_{|j| < n} (n-|j|) \exp(ij\lambda)= \dfrac{\sin^2(\frac 1 2 n\lambda)}{\sin^2(\frac 1 2 \lambda)}$. I know from Proving ...
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2answers
653 views

Is $[0,1]$ a union of family of disjoint closed intervals?

According to this question, $[0,1]$ cannot be written as union of countable disjoint closed sets, is the same true about (uncountable) family of disjoint closed intervals ?
2
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1answer
987 views

Show this function is convex.

Could someone point me in the right direction for proving the following? Given that $f:\mathbb{R}^n\rightarrow \mathbb{R}^m$ is an affine map given by $f(x)=A\mathbf{x}+\mathbf{b}$, ...
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1answer
864 views

Prove the set of which sin(nx) converges has Lebesgue measure zero (from Baby Rudin Chapter 11)

I am trying to work through Rudin. This is a question from chapter 11: Suppose that $\{n_k\}$ is an increasing sequence of positive integers and $E$ is the set of all $x$ in $(-\pi,\pi)$ at which ...
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1answer
2k views

Uniform convergence and cauchy sequence

If sequence of functions {$f_n$} converges uniformly, then {$f_n$} is a cauchy sequence. That is, it satisfies $|f_n(x)-f_m(x)| \le \epsilon$. Then if {$f_n$} is a cauchy sequence, ...
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0answers
91 views

Prove that (M,+,*) is a field

Prove that the multiplication $*:M \times M \to M$ defined by this table: * | 0 1 -------- 0 | 0 0 1 | 0 1 together with the commutative group (M,+), is a field (M,+,*). Group axioms: 1) ...
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2answers
117 views

L'Hospital's rule and Rolle's theorem

Suppose $a$ and $b$ from $\mathbb{R}$ as $a<b$ and $f$ and $g$ two continuous function on $[a;b]$ and derivable on $]a;b[$ as $\forall$ $x$ $\in$ $]a;b[$ $g{'}(x) \neq 0$. How can I prove that ...
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1answer
97 views

Hilbert basis of vector space

Let $V$ a vector space with inner product and $X\subset V$ orthonormal. Prove that exists a Hilbert basis (an orthonormal set of vectors with the property that every vector in $V$ can be written as an ...
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1answer
202 views

Is inverse use of mean value theorem right?

If we have $f$ is differentiable on $(a,b)$, and continuous on $[a,b]$, then for any $x\in (a,b)$, exists $y, z \in [a,b]$, such that $f '(x)=\dfrac{f(z)-f(y)}{z-y}$ Is this right?
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1answer
53 views

Calculate $\sum_{n=0}^{\infty} z^{n-2}/5^{n-1}$ for $0<|z|<5$ [duplicate]

Possible Duplicate: Complex series: $\sum_{n=0}^\infty\left( z^{n-2}/5^{n+1}\right)$ for $0 &amp;lt; |z| &amp;lt; 5$ I don't even know where to start. I can't think of any formulas ...
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1answer
246 views

Fact about the orthogonal complement of an subset in pre-Hilbert space

I want to show that if $X$ is a pre-Hilbert space and $A$ is a subset of $X$ with an nonempty interior, then $A^{\perp} = \{ 0 \}$. I tried to assume the contrary, then there would be an $x \ne 0$ ...
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2answers
46 views

Limit value problem

Let $f_n(x)=n^2 x(1-x^2)^n$ ($0 \le x \le 1, n=1,2,3...$) For $0<x\le1$, we have $\lim_{n→\infty}f_n(x)=0$ by the theorem. Theorem: If $p>0$ and $\alpha$ is real, then ...
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1answer
231 views

$f_m(x)=\lim_{n→\infty}(\cos m!\pi x)^{2n}$

$f_m(x)=\lim_{n→\infty}(\cos m!\pi x)^{2n}$ Define $f(x)=\lim_{m→\infty}f_m(x)$ For irrational $x$, $f_m(x)=0$ for every $m$ hence $f(x)=0$. For rational $x$, say $x=p/q$, where $p$ and $q$ are ...
3
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1answer
118 views

Closed form solution to $\{a_n\}_{n=1}^{\infty} = 1,2,2,3,3,3,…$

I had thought about this sequence (where each positive integer $n$ shows up $n$ times) the other day and think I have a closed form solution. First of all we know that the last time that $k$ shows up ...
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1answer
321 views

Continuity of a monotonically increasing function

Let $f:I\rightarrow R$ be a monotonically increasing function on an open interval.If the image of this interval is an interval then would $f$ be continuous? For the case when $f(I)$ is open then I can ...
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1answer
64 views

Let $S=(a_n)_{n=1}^{\infty}$ be a sequence in $\mathbb C$ and $S'$ the set of limits of $S$. Prove that every limit point of $S'$ is a member of $S'$

It seems obvious that a limit point of $S'$ should be a member of $S'$ but I have no idea how to even begin with a proof of this.
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1answer
334 views

damped wave equation

For $t>0$, $x$ in a compact Riemannian manifold $(M,g)$, and $a\in C^\infty(M)$, $a\geq0$, $(\partial_t^2+a\partial_t-\Delta_g)u=0$ is called the damped wave equation. My question is...why is the ...
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1answer
48 views

Implications of continuity on normed spaces

My goal for this question is to understand the implications of continuity its relationship to normed spaces. This question is based off of the notes for 18.155 available at ocw.mit.edu. Let $u$ be a ...
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2answers
167 views

Why does this integral (of the Schwarz kernel) define a holomorphic function in the unit disc?

(This may look very silly to you but I don't understand the reason that was given to me, nor do I have the knowledge to find out by myself). The domain for $z$ is the open unit disc ...
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1answer
40 views

Maximum value of a function

I need some help with the following problem. Find the maximum value of the function $f(x)=|3x^2+2ax-1|$ for $x\in[-1,1]$ if $-2 \leq a \leq 2$.
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2answers
605 views

Uniformly continuity theorem proof.

Let $f$ be continuous on [a,b]. Then $f$ is uniformly continuous on [a,b] and there exists $\delta >0$ such that $|f(s)-f(t)|<\epsilon$ if $|s-t|<\delta$. Let P={$x_0,x_1,...,x_n$} is a ...
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2answers
279 views

Show that the n-dimensional Hausdorff measure of an $n$-dimensional cube is positive and finite.

Show that the n-dimensional Hausdorff measure of an $n$-dimensional cube is positive and finite. I can easily show that if it is finite then the $n+1$ dimensional measure is $0$ and the $n-1$ ...
3
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2answers
124 views

Showing that $\lim_{x\rightarrow 0} \frac{1}{x}\int_0^x |\sin(1/y)| \mathrm{d} y \not=0$

How to show that: $$\lim_{x\rightarrow 0} \frac{1}{x}\int_0^x |\sin(1/y)| \mathrm{d} y \not=0$$ It seems like a easy example of illustrating 0 is not in the Lebesgue set of $g(x)$ where ...
2
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1answer
175 views

Example of a function f such that $f^2$ is Riemann-Stieltjes integrable on [a,b] but f is not. [duplicate]

Possible Duplicate: If $f^2$ is Riemann Integrable is $f$ always Riemann Integrable? Example of a function f such that $f^2$ is Riemann-Stieltjes integrable on [a,b] but f is not. I was ...
1
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1answer
50 views

Derivative of function where independent variable appears as a power as well.

I have $f(x)=(1+\frac{1}{x})^{x}$, I need to find the derivative of this, using the definition of derivative, and show that f is monotonically increasing. Using the definition, I have that ...
0
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2answers
781 views

Showing a function is continuous but not uniformly continuous

Let $f(x)=\frac{1}{x-3}$ on $(3,4]$. I need to show that f is continuous in this interval, but not uniformly continuous. Idea: Since $f'(x)=\frac{-1}{(x-3)^{2}}$, the derivative exists at all points ...