Articles of analysis

use of $\sum $ for uncountable indexing set

I was wondering whether it makes sense to use the $\sum $ notation for uncountable indexing sets. For example it seems to me it would not make sense to say $$ \sum_{a \in A} a \quad \text{where A is some uncountable indexing set e.g. some $A \subset \mathbb{R}$ } $$ Would it be better to […]

Sum of two periodic functions

Let $f$ and $g$ be two periodic functions over $\Bbb{R}$ with the following property: If $T$ is a period of $f$, and $S$ is a period of $g$, then $T/S$ is irrational. Conjecture: $f+g$ is not periodic. Could you give a proof or a counter example? It is easier if we assume continuity. But is […]

A sequence of real numbers such that $\lim_{n\to+\infty}|x_n-x_{n+1}|=0$ but it is not Cauchy

Give an example of a sequence $(x_n)$ of real numbers, where $\displaystyle\lim_{n\to+\infty}|x_n-x_{n+1}|=0$, but $(x_n)$ is not a Cauchy sequence

Dini's Theorem and tests for uniform convergence

Suppose $f_n$ is a sequence of functions defined a set $K$ with pointwise limit function $f$. I am confused about the following. If the following conditions are satisfied: $f_n$ is continuous on $K$ for all $n$. The pointwise limit $f$ is continuous on $K$. $K$ is a compact interval (i.e., a closed and bounded interval […]

Prove that $\int_{E}f =\lim \int_{E}f_{n}$

I’m doing exercise in Real Analysis of Folland, and got stuck on this problem. I try to use Fatou lemma but can’t come to the conclusion. Can anyone help me. I really appreciate. Consider a measure space $(X, M, \mu)$.Suppose $\{f_{n}\} \in L^{+}, f_{n} \rightarrow f$ pointwise and $\int f = \lim \int f_{n} < […]

Prove that $C^1()$ with the $C^1$- norm is a Banach Space

Consider the space of continuously differentiable functions, $$C^1([a,b]) = \{f:[a,b]\rightarrow \mathbb{R}|f,f’ \text{are continuous}\}$$ with the $C^1$-norm $$||f|| = \sup_{a\leq x\leq b}|f(x)|+\sup_{a\leq x\leq b}|f'(x)|.$$ Prove that $C^1([a,b])$ is a Banach Space. This was the proof we were given: Assuming $C^1([a,b])$ is a normed linear space all we need to show is completeness. Let $(f_n)$ be a […]

Where's the error in this $2=1$ fake proof?

This question already has an answer here: Finding the error in a proof 3 answers

Determining the action of the operator $D\left(z, \frac d{dz}\right)$

This question was motivated by a question by Tobias Kienzler and its wonderful answers. I begin as in the linked question… Using the Taylor expansion $$f(z+a) = \sum_{k=0}^\infty \frac{a^k}{k!}\frac{d^k }{dz^k}f(z)$$ one can formally express the sum as the linear operator $e^{a\frac{d}{dz}}$ to obtain $$f(z+a) = e^{a\frac{d}{dz}}f(z).$$ Other relationships were given in an answer by Tom […]

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 continuously differentiable, under what conditions can we guarantee we can find a polynomial $p$ such that $p$ approximates arbitrarily well $f$ and the partial derivatives approximate of $p$ […]

Show that the norm of the multiplication operator $M_f$ on $L^2$ is $\|f\|_\infty$

I’m having some (hopefully small) issues computing the norm of an operator. Firstly, the problem, For $f\in L^\infty[0,1]$, define $M_f: L^2[0,1]\to L^2[0,1]$ by $M_f(g)(x) = f(x)g(x)$. Show that $M_f$ is a bounded linear operator and $\|M_f\| = \|f\|_\infty$. What I’ve done so far: To see that $M_f$ is linear, let $g_1,g_2\in L^2[0,1]$ and $\lambda\in\mathbb{R}$. Then […]