Sequence of bounded functions or pointwise bounded

Question

We have sequence of function $\{f_n(x)\}$ defined on set $E$ and $n\in \mathbb{N}$. What does mean sequence of bounded functions? Is it pointwise bounded?

Definition: We say that $\{f_n\}$ is pointwise bounded on $E$ if the sequence $\{f_n(x)\}$ is bounded for every $x\in E$, that is, if there exists a finite-valued function $\phi$ defined on $E$ such that $$|f_n(x)|<\phi(x) \quad (x\in E, n\in \mathbb{N}).$$

Can anyone explain to me please?

just to add.why it is NOT pointwise bounded because once we know that it has uniform convergence it implies pointwise boundedness — manifold, Nov 20 '19 at 12:40

score 7 · Accepted Answer · answered Nov 26 '15 at 20:04

7

A bounded function is a function whose range is contained in a finite interval. A sequence of bounded functions has the property that each of its individual functions $f_n$ has such a limited range (but the overall range of the collection may still be infinitely large).

The sequence is pointwise bounded if for every $x$ separately the sequence $f_n(x)$ is contained in a finite interval. That is a different condition.

answered Nov 26 '15 at 20:04

Justpassingby

10,029

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So sequence of bounded functions ${f_n(x)}$ means that $|f_n(x)|<M_n$ for any $n$. Right? – RFZ Nov 26 '15 at 20:06
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Exactly. And pointwise bounded sequence means that $|f_n(x)|<m_x$ for any $x.$ – Justpassingby Nov 26 '15 at 20:11

Sequence of bounded functions or pointwise bounded

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