Examples
- For some fixed g ∈ C(R), define the linear operator T by
- That the operator T is indeed compact follows from the Ascoli theorem.
- More generally, if Ω is any domain in Rn and the integral kernel k : Ω × Ω → R is a Hilbert—Schmidt kernel, then the operator T on L2(Ω; R) defined by
- is a compact operator.
- By Riesz's lemma, the identity operator is a compact operator if and only if the space is finite dimensional.
Read more about this topic: Compact Operator
Famous quotes containing the word examples:
“No rules exist, and examples are simply life-savers answering the appeals of rules making vain attempts to exist.”
—André Breton (18961966)
“It is hardly to be believed how spiritual reflections when mixed with a little physics can hold peoples attention and give them a livelier idea of God than do the often ill-applied examples of his wrath.”
—G.C. (Georg Christoph)
“In the examples that I here bring in of what I have [read], heard, done or said, I have refrained from daring to alter even the smallest and most indifferent circumstances. My conscience falsifies not an iota; for my knowledge I cannot answer.”
—Michel de Montaigne (15331592)