Helly's selection theorem

From Wikipedia, the free encyclopedia

In mathematics, Helly's selection theorem states that a sequence of functions that is locally of bounded total variation and uniformly bounded at a point has a convergent subsequence. In other words, it is a compactness theorem for the space $BV loc$ .

It is named for the Austrian mathematician Eduard Helly.

[edit] Statement of the theorem

Let $U$ be an open subset of the real line and let $\scriptstyle f_{n} : U \to \mathbb{R}$ , $\scriptstyle n \in \mathbb{N}$ , be a sequence of functions. Suppose that

$(f n)$ has bounded total variation on any $W$ that is compactly embedded in $U$ . That is, for all open sets $\scriptstyle W \subseteq U$ with $\scriptstyle\bar{W} \subsetneq U$ compact,

$\sup_{n \in \mathbb{N}} \left( \left\| f_{n} \right\|_{L^{1} (W)} + \left\| \frac{\mathrm{d} f_{n}}{\mathrm{d} t} \right\|_{L^{1} (W)} \right) < + \infty,$

where the derivative is taken in the sense of tempered distributions.

$(f n)$ is uniformly bounded at a point. That is, for some $\scriptstyle t \in U$ , $\scriptstyle\{ f_{n} (t) | n \in \mathbb{N} \} \subsetneq \mathbb{R}$ is a bounded set.

Then there exists a subsequence $\scriptstyle(f_{n_{k}})_{k = 1}^{\infty}$ and a function $\scriptstyle f : U \to \mathbb{R}$ locally of bounded variation such that

$\scriptstyle f_{n_{k}}$ converges to $f$ pointwise;
$\scriptstyle f_{n_{k}}$ converges to $f$ in $\scriptstyle L_{\mathrm{loc}}^{1} (U; \mathbb{R})$ (see locally integrable function);
and, for $W$ compactly embedded in $U$ as above,

$\left\| \frac{\mathrm{d} f}{\mathrm{d} t} \right\|_{L^{1} (W)} \leq \liminf_{k \to \infty} \left\| \frac{\mathrm{d} f_{n}}{\mathrm{d} t} \right\|_{L^{1} (W)}.$

Categories: Mathematical analysis | Mathematical theorems

Helly's selection theorem

From Wikipedia, the free encyclopedia

[edit] Statement of the theorem

[edit] See also

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