= Compact embedding
{title2=$X\hookrightarrow\hookrightarrow Y$}
A <compact embedding> of <Banach spaces> $X,Y$ is an injective <continuous linear map> $i:X\to Y$ that sends every <bounded set> to a <relatively compact subset>. Equivalently, every bounded <sequence> in $X$ has a <subsequence> converging in $Y$. One identifies $X$ with its image when $i$ is the natural inclusion. The <Rellich-Kondrachov theorem> supplies many <compact embeddings> between <Sobolev spaces> and <Lp spaces> on bounded <domains>.
Back to article page
