Sobolev Space < Hilbert Space < Banach Space

最近在看有限元的书,这几个概念已经让我快疯掉了。

Sobolev space is a vector space of functions equipped with a norm that is a combination of Lp norms of the function itself as well as its derivatives up to a given order.

Hilbert space H is a real or complex inner product space that is also a complete metric space with respect to the distance function induced by the inner product.

Banach spaces are defined as complete normed vector spaces. This means that a Banach space is a vector space V over the real or complex numbers with a norm ||·|| such that every Cauchy sequence (with respect to the metric d(x, y) = ||x − y||) in V has a limit in V.

先把概念抄在这儿吧,看懂再说。

用有限元法求解偏微分方程,解存在于Sobolev空间,主要是这里有个弱解的问题。在弱解的情况里,对函数的光滑性要求降低,不必要二阶连续可导,只要平方可积即可。

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