# Tagged Questions

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### Can all functions be expressed in terms of elementary functions?

After being introduced to the non-elementary function through an attempt to evaluate $\int x \tan (x)$, an interesting question occurred to me: Can the non-elementary functions be decomposed to ...
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### Is this construction already a contradiction

Let's say we have $\ell^p$ for $p>2$ and $\ell^2 \subsetneq \ell^p$. Let $U$ be a closed subspace of $\ell^2$. Is it possible that $U$ is isomorphic to $\ell^p$ as a Banach space?- I hardly think ...
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### $C^\infty$ approximations of $f(r) = |r|^{m-1}r$

Consider $f(r) = |r|^{m-1}r$ where $m \geq 1$. Is it possible to find $C^\infty$ functions $f_n$, such that $f_n \to f$ uniformly on compact subsets of $\mathbb{R}$, $f_n' \to f'$ uniformly on ...
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### Transfer vector space properties to dual space

I am curious about this here (Actually, I don't know if my assumptions are true or not) a) Let $X$ be a Banach space that is isomorphic to $Y$, then $X^*$ is also isomorphic to $Y^*$. I sketched a ...
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### How to find maximum/minimum of $y=\frac{x(x^2-x+2)}{x^2-9}$?

How can I find minimum and maximum of $y=\frac{x(x^2-x+2)}{x^2-9}$? In other words, points, where $y'=0$. My current steps are: $$y=\frac{x(x^2-x+2)}{x^2-9}=x-1+\frac{11x-9}{x^2-9}$$ ...
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### $l^r \subset l^p$ and is it even a subspace

It is true that for $r<p$ and $r,p \in [1,\infty)$ we have that $l^r \subset l^p$. Is it true that $l^r$ cannot be isomorphic to a subspace of $l^p$?
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### Is $l^p$ closed in $L^p$?

Let's assume we have a subspace $X$ of $L^p$ and we know that $X \cong l^p$(this should just mean isomorph no isometry is assumed here). Can we infer from this that $X$ is closed? I have just read a ...
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### Basic sequence- what is so special about it?

Let $(x_n)$ be a Schauder basis of a vector space $X$. This means that the $span(x_n)$ is dense in $X$, right? Then wikipedia introduces the notion of a $\textbf{basic sequence}$ when $(x_n)$ is a ...
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### Biorthogonal functions in $L^p$

I asked one question that is already answered: 1.) I have a question about Lemma 9.5 on page 93/94 reference. It's about the part of the proof where the sequence of $(g_n^*)$ are introduced. I don't ...
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### Perturbation theory PDEs

I have the solution of a PDE of the form: $$\Delta \Psi(r,\theta, \phi) = k \Psi(r,\theta,\phi)$$ on a set $\mathbb{R}^3 \backslash B(0,R)$. Hence, the actual solution is known there! Regarding ...
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### old prelim exam question

In studying for my prelims, I can't quite come up with a solution to the following problem, from a few years ago. It has the feel of a "write the correct thing down, and it's 3 lines" problem. ...
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### Riesz Fischer theorem?

I was wondering about the following: I read that Fischer-Riesz says that $L^2([0,1])$ is isomorphic to $l^2(\mathbb{N})$. Now it is obvious, that this should not depent on the fact which compact ...
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### When integration by parts applies?

I think integration by parts is really useful, but I don't know when integration by parts applies. Take the following proposition as an example: Let $\Omega \subset \mathbb{R}^n$ be an open set, ...
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### Limit of a hyperpower function

i have a question regarding this class of equations: Let $\gamma(x)=x^x$ Let $\Psi_n(x)=\underbrace{\gamma(x)\circ\gamma(x)\circ\gamma(x)}_n$, such that $\Psi_1(x)=\gamma(x)$ and ...
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### A inequality of calculus [duplicate]

Let $f \in C^2[a,b]$ and $f(a) = f(b) = 0$, $f'(a) = 1$,$f'(b) = 0$, prove that $$\int_a^b|f''(x)|^2\,dx \geq \frac{4}{b-a}$$ Remark: This question is in the book functional analysis of Peking ...
In Quantum Mechanics one often looks at self-adjoint(unbounded and closed) linear operators $A,B$ that are defined dense on $L^2$. My question is: Is it true that if we have $[A,B]=0$, where $[,]$ is ...