Fréchet–Kolmogorov Theorem

In functional analysis, the Fréchet–Kolmogorov theorem (the names of Riesz or Weil are sometimes added as well) gives a necessary and sufficient condition for a set of functions to be relatively compact in an ''L''^''p'' space. It can be thought of as an ''L''^''p'' version of the Arzelà–Ascoli theorem, from which it can be deduced. The theorem is named after Maurice René Fréchet and Andrey Kolmogorov.

Statement

Let $B$ be a subset of $L^p(\mathbb^n)$ with $p\in$relatively compact if and only if the following properties hold:

#(Equicontinuous) $\lim_\Vert\tau_h f-f\Vert_ = 0$ uniformly on $B$.
#(Equitight) $\lim_\int_\left, f\^p=0$ uniformly on $B$.

The first property can be stated as $\forall \varepsilon >0 \, \, \exists \delta >0$ such that $\Vert\tau_h f-f\Vert_ < \varepsilon \, \, \forall f \in B, \forall h$ with $, h, <\delta .$

Usually, the Fréchet–Kolmogorov theorem is formulated with the extra assumption that $B$ is bounded (i.e., $\Vert f\Vert_<\infty$ uniformly on $B$). However, it has been shown that equitightness and equicontinuity imply this property.

Special case

For a subset $B$ of $L^p(\Omega)$, where $\Omega$ is a bounded subset of $\mathbb^n$, the condition of equitightness is not needed. Hence, a necessary and sufficient condition for $B$ to be relatively compact is that the property of equicontinuity holds. However, this property must be interpreted with care as the below example shows.

Examples

Existence of solutions of a PDE

Let $(u_\epsilon)_\epsilon$ be a sequence of solutions of the viscous Burgers equation posed in $\mathbb\times(0,T)$:

:$\frac + \frac\frac = \epsilon\Delta u, \quad u(x,0) = u_0(x),$

with $u_0$ smooth enough. If the solutions $(u_\epsilon)_\epsilon$ enjoy the $L^1$-contraction and $L^\infty$-bound properties, we will show existence of solutions of the inviscid Burgers equation

:$\frac + \frac\frac = 0, \quad u(x,0) = u_0(x).$

The first property can be stated as follows: If $u,v$ are solutions of the Burgers equation with $u_0,v_0$ as initial data, then

:$\int_, u(x,t)-v(x,t), dx\leq \int_, u_0(x)-v_0(x), dx.$

The second property simply means that $\Vert u(\cdot,t)\Vert_\leq \Vert u_0\Vert_$.

Now, let $K\subset\mathbb\times(0,T)$ be any compact set, and define

:$w_\epsilon(x,t):=u_\epsilon(x,t)\mathbf_K(x,t),$

where $\mathbf_K$ is $1$ on the set $K$ and 0 otherwise. Automatically, $B:=\\subset L^1(\mathbb^2)$ since

:$\int_, w_\epsilon(x,t), dx dt= \int_, u_\epsilon(x,t)\mathbf_K(x,t), dx dt\leq \Vert u_0\Vert_, K, <\infty.$

Equicontinuity is a consequence of the $L^1$-contraction since $u_\epsilon(x-h,t)$ is a solution of the Burgers equation with $u_0(x-h)$ as initial data and since the $L^\infty$-bound holds: We have that

:$\Vert w_\epsilon(\cdot-h,\cdot-h)-w_\epsilon\Vert_\leq \Vert w_\epsilon(\cdot-h,\cdot-h)-w_\epsilon(\cdot,\cdot-h)\Vert_+\Vert w_\epsilon(\cdot,\cdot-h)-w_\epsilon\Vert_.$

We continue by considering

:$\begin &\Vert w_\epsilon(\cdot-h,\cdot-h)-w_\epsilon(\cdot,\cdot-h)\Vert_\\ &\leq \Vert (u_\epsilon(\cdot-h,\cdot-h)-u_\epsilon(\cdot,\cdot-h))\mathbf_K(\cdot-h,\cdot-h)\Vert_+\Vert u_\epsilon(\cdot,\cdot-h)(\mathbf_K(\cdot-h,\cdot-h)-\mathbf_K(\cdot,\cdot-h)\Vert_. \end$

The first term on the right-hand side satisfies

:$\Vert (u_\epsilon(\cdot-h,\cdot-h)-u_\epsilon(\cdot,\cdot-h))\mathbf_K(\cdot-h,\cdot-h)\Vert_\leq T\Vert u_0(\cdot-h)-u_0\Vert_$

by a change of variable and the $L^1$-contraction. The second term satisfies

:$\Vert u_\epsilon(\cdot,\cdot-h)(\mathbf_K(\cdot-h,\cdot-h)-\mathbf_K(\cdot,\cdot-h))\Vert_\leq \Vert u_0\Vert_\Vert \mathbf_K(\cdot-h,\cdot)-\mathbf_K\Vert_$
by a change of variable and the $L^\infty$-bound. Moreover,

:$\Vert w_\epsilon(\cdot,\cdot-h)-w_\epsilon\Vert_\leq \Vert (u_\epsilon(\cdot,\cdot-h)-u_\epsilon)\mathbf_K(\cdot,\cdot-h)\Vert_+\Vert u_\epsilon(\mathbf_K(\cdot,\cdot-h)-\mathbf_K)\Vert_.$

Both terms can be estimated as before when noticing that the time equicontinuity follows again by the $L^1$-contraction. The continuity of the translation mapping in $L^1$ then gives equicontinuity uniformly on $B$.

Equitightness holds by definition of $(w_\epsilon)_\epsilon$ by taking $r$ big enough.

Hence, $B$ is relatively compact in $L^1(\mathbb^2)$, and then there is a convergent subsequence of $(u_\epsilon)_\epsilon$ in $L^1(K)$. By a covering argument, the last convergence is in $L_^1(\mathbb\times(0,T))$.

To conclude existence, it remains to check that the limit function, as $\epsilon\to0^+$, of a subsequence of $(u_\epsilon)_\epsilon$ satisfies

:$\frac + \frac\frac = 0, \quad u(x,0) = u_0(x).$

See also

*Arzelà–Ascoli theorem
* Helly's selection theorem
* Rellich–Kondrachov theorem

References

Literature

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{{DEFAULTSORT:Frechet-Kolmogorov Theorem
Theorems in functional analysis
Compactness theorems