Many problems arising in science and engineering call for the solving of the Euler-Lagrange equations of functionals. Thus, solving the Euler-Lagrange equations is tantamount to finding critical points of the corresponding functional. An idea that has been very successful is to find appropriate sets that separate the functional. This method is called linking. Two sets $A,B$ are said to separate a functional G if the supremum of G on A is less than or equal to the infimum of G on B. Two sets of the space are said to link if they produce a critical sequence whenever they separate a functional. If the critical sequence has a convergent sub-sequence, then that produces a critical point. Finding sets that separate a functional is quite easy, but determining whether or not the sets link is quite another story. It appears that the only way we can check to see if two sets link, is to require that one of them be contained in a finite-dimensional subspace. The reason is that in order to verify the definition, we need to invoke the Brouwer fixed point theorem. Our aim is to find a counterpart of linking that holds true when both sets are infinite dimensional. We adjust our definitions to accommodate infinite dimensions. These definitions reduced to the usual definitions when one set is finite dimensional. In order to prove the corresponding theorems, we make adjustments to the topology of the space and introduce infinite dimensional splitting. This allows us to use a form of compactness on infinite dimensional subspaces which does not exist under the usual topology. We lose the Brouwer index, but we are able to replace it with the Leray-Schauder index. We carry out the details in Sections 5 and 6. In Section 7 we solve a system of equations which require infinite dimensional splitting.

科学和工程学中出现的许多问题都要求求解泛函的Euler-Lagrange方程。因此，求解欧拉-拉格朗日方程等于找到相应泛函的临界点。一个非常成功的想法是找到分离功能的适当集合。此方法称为链接。两套$\mathrm{一个}，乙$被说成分离的功能ģ如果上确界ģ上甲小于或等于的下确界ģ上乙。如果两套空间在分隔功能时会产生关键序列，则称为链接。如果关键序列具有收敛的子序列，则将产生一个关键点。查找分离功能的集合非常容易，但是确定集合链接是否完全是另一回事。看来，我们可以查看是否有两个集合链接的唯一方法是要求其中一个集合包含在有限维子空间中。原因是为了验证定义，我们需要调用Brouwer不动点定理。我们的目标是找到当两个集合都是无穷维时也适用的链接的对应项。我们调整定义以适应无限的尺寸。当一组是有限维时，这些定义简化为通常的定义。为了证明相应的定理，我们对空间的拓扑结构进行了调整，并引入了无限维分裂。这使我们能够对通常的拓扑结构中不存在的无限维子空间使用紧凑形式。我们失去了Brouwer指数，但是可以用Leray-Schauder指数代替它。我们将在第5节和第6节中进行详细介绍。在第7节中，我们解决了需要无限维分解的方程组。