 Let ("X", "T") be a topological space, and let "V" and "W" be subsets of "X".We say that "V" is compactly embedded in "W", and write "V" ⊂⊂ "W", if

Let "X" and "Y" be two normed vector spaces with norms ||•||"X" and ||•||"Y" respectively, and suppose that "X" ⊆ "Y".We say that "X" is compactly embedded in "Y", and write "X" ⊂⊂ "Y", if

If "Y" is a Banach space, an equivalent definition is that the embedding operator (the identity) "i"