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).

learn more… | top users | synonyms (1)

0
votes
1answer
56 views

Does it matter in functional analysis whether we know (something about) a basis?

Let's look at a few spaces in functional analysis: $(L^p,C^n([0,1]), l^p,c,c_0,d)$ I actually only know the basis of one of these spaces. Which is the one that belongs to d, given by the unit ...
3
votes
2answers
145 views

Show $S(t) =\sum_{n=-\infty}^\infty\sin{(n^2t^2)}e^{-tn^2}$ is $O(t^p)$ at zero

An old qualifying exam problem: For $t>0$, define $$S(t) =\sum_{n=-\infty}^\infty\sin{(n^2t^2)}e^{-tn^2}.$$ Show that $S(t) = C t^p + o(t^p)$ as $t\to 0$ . Find $C$ and $p$. There are a couple of ...
0
votes
1answer
126 views

Prove that subsequences of a sequence with infinite range contained in a compact set X converges to a point in X.

I have a question about a step regarding the following proof from Rudin theorem 3.6a), Proof: Let E be the range of the sequence ${p_n}$ since E is a infinite subset of a compact set it has a limit ...
0
votes
2answers
74 views

Showing that a sequence converges/diverges

I've managed to prove that (a) converges to 0 and (d) diverges. However I am stuck on (b) and (c). For (b), I assumed that it converges to $\dfrac{3}{2}$ Then did $|a_n -L|$and got to ...
3
votes
1answer
82 views

Interesting inequality $\|F\|_p\le \frac{\pi}{\sin(\pi/p)}\|f\|_p$ over $L^p$

Consider the function $$F(x)=\int_0^\infty \frac{f(y)}{x+y} \, dy, \quad0<x<\infty$$ Prove that if $1<p<\infty$, $$\|F\|_p\le \frac{\pi}{\sin(\pi/p)}\|f\|_p$$ and show that the constant ...
0
votes
1answer
67 views

Give an example of $f\in L^1$, $g\in L^{\infty}$, such that $f*g\notin C_0$ (meaning $\lim_{|x|\to\infty}(f*g)(x)\neq0$)

Give an example of $f\in L^1$, $g\in L^{\infty}$, such that $f*g\notin C_0$ (meaning $\lim_{|x|\to\infty}(f*g)(x)\neq0$) Here's a theorem from my real analysis book: Assume $1\le p\le \infty$ and ...
1
vote
2answers
381 views

Limit of functions and the Binomial Theorem

If $n \geq 2$ is an integer, show $n^{1/n} = 1 + h$; where $h \leq \sqrt{ \dfrac{2}{n-1}}$ Then Deduce that: $\lim\limits_{n \to \infty} n^{1/n} = 1$ Hint: Since $n>1$, $n^{1/n}>1$. So, ...
2
votes
1answer
312 views

Show using dominated convergence that summation and differentiation can be interchanged.

Problem: Define $\{p_k: k \ge 0 \}$ as a probability mass function on a discrete random variable X taking values on $\{0,1,...\}$ then define the generating function $P(x) = E(x^X) = ...
2
votes
4answers
466 views

Limit of $x_1=1, x_{n+1}=x_n+\frac1{x_n^2}$

Given that $x_1 = 1$ and $x_{n+1}=x_{n}+ 1/x_{n}^{2}$. Find the limit of the sequence. Let $ c $ be the limit of the sequence, then $ c=c+\frac{1}{c^2} $, that means $ \frac{1}{c^2}=0 $. That ...
0
votes
1answer
56 views

Continuity of functions from complex numbers

i have a question about continuity. Suppose i have a function from $\Bbb{C}$ into a Banachalgebra $A$ for example $r\mapsto exp(ra)$ for a fixed $a\in A$. Do we have to prove continuity by ...
1
vote
1answer
69 views

Find the sum of the series

For any integer $n$ define $k(n) = \frac{n^7}{7} + \frac{n^3}{3} + \frac{11n}{21} + 1$ and $$f(n) = 0 \text{if $k(n)$ is an integer ; $\frac{1}{n^2}$ if $k(n)$ is not an integer } $$ Find $\sum_{n = ...
0
votes
1answer
31 views

Magnitude of Fourier coefficients when $||f||_2 \leq 1$

Let $f \in L_2[- \pi, \pi] $ so that $||f||_2 \leq 1$. Can I say anything about $f$'s Fourier coefficients' magnitude without assuming anything else about $f$? To be more accurate: The squared norm ...
5
votes
2answers
1k views

Is there any handwavy argument that shows that $\int_{-\infty}^{\infty} e^{-ikx} dk = 2\pi \delta(x)$?

It should not be a good argument but rather a short one and one that convinces a physicist ( so no need for mathematical rigor ) that shows that $\int_{-\infty}^{\infty} e^{-ikx} dk = 2\pi \delta(x)$ ...
3
votes
1answer
325 views

Zeros of complex function sequence (Application of Rouche's Theorem).

For a given sequence of complex functions: $\phi_n(z)= 1+\frac1n-z-e^{-z}$; here $z\in${$z| Rez>0$}. I want to prove that : (1). $\phi_n $ has a unique zero $z_n$ in the half plane. (i.e. there ...
1
vote
1answer
258 views

Is it possible to generalize the Mean value theorem for integral not on compact set?

I wonder it is possible to extend the mean value theorem not on compactness. In more detail, Let $f : A \rightarrow \mathbb{R}$ be continuous on $A \subset \mathbb{R}^n$. The mean value theorem for ...
0
votes
1answer
194 views

Why is this not a space-filling curve?

From Wikipedia, a space-filling curve is a curve (i.e. a continuous function whose domain is the unit interval $[0,1]$) whose range contains the entire 2-dimensional unit square. Many examples of ...
3
votes
1answer
40 views

Question on series

Suppose $ a_i $ be a sequence of positive real numbers such that $ \sum_{i=1}^{\infty}a_i < \infty $. Is it true that $ \sum_{i=1}^{\infty} \log(i) \cdot a_i < \infty $? Thanks
1
vote
2answers
52 views

Limit of Fuctions

Let $f(x)= \left \{ \begin{array}{cc} x & x\in \mathbb{Q}\\ 0 & \,\,\,\,\,\,x\in \mathbb{R}\setminus\Bbb{Q} & \end{array} \right . $ Determine all $a \in \mathbb{R}$ for which ...
8
votes
2answers
180 views

Prove that $|f''(\xi)|\geqslant\frac{4|f(a)-f(b)|}{(b-a)^2}$

Let ${\rm f}:\left[a, b\right]\to\mathbb{R}$ be twice differentiable, and suppose $$\lim_{x\to a^{+}} \frac{{\rm f}\left(x\right) - {\rm f}\left(a\right)}{x - a} = \lim_{x\to b^{-}}\frac{{\rm ...
2
votes
1answer
26 views

Existence of certain set

Problem: Let $0<a<1$. Let $\lambda$ be the Lebesgue measure on $\mathbb{R}$. Show that: (i) There exists a closed set $A\subseteq[0,1]$, which does not contain any non-empty open sets, such ...
4
votes
1answer
101 views

Non isolated minimum

Consider the $C^2$ function $F:\mathbb{R}^k \rightarrow \mathbb{R}^k$ is it possible for it to be such that $x_n$ is a strict local minimizer for all $n$ and $x_n \rightarrow x$ where $x$ is a strict ...
5
votes
2answers
68 views

I'm stuck with this.. (number 9,6 and 3)

Hello guys/girls I was bored and I just played around with math. I am stuck and it's about raised numbers. (9, 6 and 3) So this is how you calculate it. (same method for all numbers) Raise 3, 6 and ...
2
votes
1answer
335 views

Convergence of a integral - heat Kernel and dirac delta function

Consider $\varphi \in S(R^n)$ (space of rapidly decreasing functions). Consider the heat kernel $$ K_t(x) = \displaystyle\frac{1}{{(4\pi t)}^{n/2}} \displaystyle e^{- \displaystyle\frac{|x|^2}{4t}}, ...
1
vote
1answer
58 views

Why is it that the vector space of all derivations has the basis $\partial/\partial{x_1} \ldots \partial/\partial{x_n}$?

So I have seen stated that two definitions of tangent spaces (w.r.t manifolds) are equivalent. But I am having some difficulty proving they are indeed equivalent. It looks like it boils down to me ...
1
vote
1answer
52 views

When is something coordinate independent

One defines a critical point of a function $f$ to be a point where all its partial derivatives vanish. The critical points fall into two classes, degenerate and non-degenerate ones. A critical point ...
-4
votes
1answer
82 views

Proving various properties of metric spaces

Suppose that $p_1$ and $p_2$ are metrics on $M$. Prove that the following are also metrics: (a) $p = p_1 + p_2$ define $p_1(x,y) = |x-y|$ define $p_2(a,b) = |a-b|. p = p_1+p_2 = |x-y|+|a-b|$. But I ...
1
vote
1answer
108 views

Dirac Delta — Symmetry

I had a curiosity question rise up in the middle of the night regarding the behavior of the Dirac Delta. Because it's not a function per-se, I am not sure how a concept like "integration" symmetry ...
1
vote
1answer
54 views

Any predetermined sequence in the decimal expansion of an irrational number

I came up with this question in a random math discussion with my friend. I am wondering if one can always find a predetermined sequence of numbers, such as 123456, 33333, in the decimal expansion of a ...
0
votes
2answers
86 views

proof of differentiatiable function

prove that x^(1/3) is differentiable at a with f(a)'=((a^(1/3))^-2)/3 for all a not equal to 0. I tried a epsilon-delta proof with limes theorem, and or that does not work or I am making somewhere ...
5
votes
1answer
192 views

looking for a diffeomorphism (not C1)

Let $f\colon\mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ diffeomorphism with $f(B[0,1])\subset B[0,1]$ and $| \det f^{\prime}(x) |<1/2$ for all $x\in B[0,1]$ then for every continuous function ...
2
votes
2answers
132 views

J-measurable sets and functions of class $C^1$

If $f:\mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ is $C^1$ class and $det f^{\prime}(0)=0$ show that, when $r\rightarrow 0$ $$\dfrac{Vol(f(B[0,r]))}{ Vol(B[0,r])} \rightarrow 0$$ where $Vol(X)$ is ...
7
votes
1answer
558 views

Dini's continuity vs Holder continuity

(listed items are just the definitions, you can skip to "Clearly" if you are familiar with them) Let $E \subset \mathbb{R}^N$ and let $f \colon E \to \mathbb{R}.$ The modulus of continuity of $f$ is ...
0
votes
0answers
40 views

No Subset of a Path is open

Let $\gamma:[a,b]\subset\mathbb{R}\rightarrow\mathbb{C}$ be continously differentiable. Prove that no subset $A\subset \gamma([a,b])$ is open in $\mathbb{C}$. And would this also be true if $\gamma$ ...
0
votes
1answer
157 views

Which of these norms are equivalent to the canonical one

Regarding the space of continuously differentiable functions $C^1([0,1])$, I am wondering which of these norms are equivalent to the norm $||x||= ||x||_{\infty} + ||x'||_{\infty}$. The candidates are ...
0
votes
1answer
137 views

Show that $\alpha t-\log(t)$ has compact level set.

Let $\mathbb{R}_{++}=\{ x \in \mathbb{R} \mid x>0\}$. I need to show that for $t\in \mathbb{R}_{++}$: $t\mapsto \alpha t -\log(t)$ has a compact level set for $\alpha \in \mathbb{R}_{++}$, where ...
0
votes
2answers
490 views

Contraction Mapping, why constant and weak inequality?

From Wikipedia, a contraction mapping is a function $f: M \rightarrow M$ on a metric space $(M,d)$ such that there exists a nonnegative real number $k<1$ such that for all $x,y\in M$, $$ d ...
3
votes
1answer
533 views

On finding polynomials that approximate a function and its derivative (extensions of Stone-Weierstrass?)

The Stone-Weierstrass Theorem tells us that we can approximate any continuous $f:\mathbb{R}^n\to\mathbb{R}$ arbitrary well on a compact subset of $\mathbb{R}^n$ by some polynomial. Suppose that $f$ is ...
2
votes
1answer
143 views

Why a uniform limit of a sequence of bounded derivative is a derivative?

Assume that a sequence $(f_n)$ of functions $f_n:[0,1] \rightarrow \mathbb R$ is uniformly convergent to a function $f:[0,1]\rightarrow \mathbb R$. Moreover let (for each $n\in \mathbb N$) $f_n$ be ...
1
vote
1answer
103 views

How do I draw a diagram for a function space?

If one considers a single function, then one can just draw its diagram as a Cartesian product. So it's relatively easy to contribute one's intuition to an argument. However, when it is a function ...
1
vote
3answers
139 views

Calculating the integral expression $\int_0^{\infty}\lambda^{t-1}e^{-\lambda z}d\lambda$ for complex-valued z

The problem is computing the integral expression $$f(t)=\int_0^{\infty}\lambda^{t-1}e^{-\lambda z}d\lambda$$ where z is a complex variable with $Re(z)> 0$ and t is a real variable. Is it correct to ...
3
votes
1answer
109 views

Lower bound on a number theoretic function

Let $n$ be a positive odd integer, let $$n_j = \Bigl\{\frac{n}{2^{j+1}}\Bigr\}\,,$$ where $\{x\}$ denotes the fractional part of $x$, and finally let $k = \lceil \log_2 n\rceil$. Consider the ...
3
votes
1answer
89 views

Example of two series with certain properties?

Find 2 series $\sum a_k$ and $\sum b_k$ such that $\sum b_k$ converges conditionally, $\dfrac{a_k}{b_k} \rightarrow 1$ as $k \rightarrow \infty$, and $\sum a_k$ diverges. Can someone give me a hint ...
2
votes
2answers
455 views

Can we cover a closed interval with open sets?

This might be extremely obvious but a proof such as this one show a "cover" of [0,2] to be: the collection of open sets: $\{ (\frac{1}{n},2) \mid n \text{ is a positive integer} \}$ This is ...
2
votes
0answers
37 views

If every continous function $f:X\rightarrow \mathbb{R}$ has its image as an interval, then $X$ is connected.

To prove by contradiction, suppose $X$ is not connected. Then $X$ can be written as a union of two nonempty disjoint open set $U$ and $V$. Then can I assume that there exists a one to one function ...
1
vote
1answer
57 views

Is this function continuous at $(0, 0)$?

Suppose a function $f(x, y)$ is defined as follows like this: $f(x, y)=\frac{xy^3}{x^3+y^4}$ when $(x, y)\neq (0, 0)$ and $f(x, y)=(0, 0)$ when $(x, y)=(0, 0)$. Is this function continuous at $(0, ...
1
vote
0answers
65 views

Nilpotents of order 2 are dense in the strong operator topology

I need some help with this homework problem. (The Hilbert space in question is $\ell^2(\mathbb{N})$.) I let $T \in \mathcal{B}(\mathcal{H})$ and looked at a basic nbhd in the strong operator topology ...
0
votes
2answers
62 views

Prove that $\lim x_k = a \iff \lim \langle x_k, y \rangle = \langle a, y \rangle$ $\forall$ $y \in \mathbb{R}^n$.

Could someone help me prove this real analysis theorem? Prove that $\lim x_k = a \iff \lim \langle x_k, y \rangle = \langle a, y \rangle$ $\forall$ $y \in \mathbb{R}^n$.
0
votes
1answer
26 views

Help Understanding: Closed Subspace of Compact Space is Compact on ProofWiki

http://www.proofwiki.org/wiki/Closed_Subspace_of_Compact_Space_is_Compact ProofWiki provides the following proof that a closed subspace of a compact space is compact: Let $T$ be a compact space. Let ...
2
votes
1answer
195 views

Weak implicit function theorem. Is my proof alright?

I want to prove that if we have $ f \in C(A \times U,Y)$, where $x_0 \in U$, $\lambda_0 \in A$ and A and U are open sets in Banach spaces and Y is a Banach space too and we have that: ...
2
votes
1answer
71 views

Prove or disprove a set $F$ is closed.

This is an example in my book that talks about $F$ being precompact; Let $F$ be the subset of $C([0,1])$ that consists of functions $f$ of the form $$f(x) = \sum_{n=1}^{\infty}a_n\sin(n\pi x) ...