6k views

930 views

### if f is continuous and absolutely integrable $\lim\limits_{x\to\infty}f(x)=0$

Prove that if f(x) is continuous and absolutely integrable on $[a,\infty)$ then $\lim\limits_{x\to\infty}f(x)=0$. I tried proving it in the following way: First we need to prove the existence of the ...
275 views

### Assume that $f$ is uniformly continuous. Prove that $\lim_{x→∞} f(x) = 0.$

I think the following question is probably fairly easy but can't think of an easy way of proving it. Some help would be awesome. This question comes from an old qual. Thanks. Let $f$ be an ...
304 views

### Essential Selfadjointness of Quantum Harmonic Oscillator Hamiltonian

The Hamiltonian for the Quantum Harmonic Oscillator is (disregarding constants) the Hermite operator $$Hf = -f''+x^{2}f,$$ where $\mathcal{D}(H)$ consists of all twice absolutely ...
451 views

### Behaviour of $L^2$ functions at infinity [duplicate]

Is it possible to prove that if $f\in L^2(\mathbb {R})$ then $\exists\lim_{x\to\pm\infty}\lvert f\rvert^2$ and $\lim_{x\to\pm\infty}\lvert f\rvert^2=0$? If not, is it easy to find a counterexample?
### When does $f\in L^1$ vanish?
I am asking about the sort-of converse to this question: under what additional conditions on $f:\mathbb{R}_+\rightarrow\mathbb{R}$ does the following hold?  \int_{\mathbb{R}_+}f<\infty\implies \...
### Suppose $f : X \to Y$ is a (continuous) bounded map.Does this implies that $f$ is uniformly continuous?
It's well known that if $f : \bf (X,d) \to \bf (Y,e)$ is a uniformly continuous function then $f$ maps bounded set to bounded set.Does the converse hold ? More Precisely, Suppose $f : X \to Y$ ...