The Set of Invertible Operators Is Open
A corollary is that the set of invertible operators between two Banach spaces B and B' is open in the topology induced by the operator norm. Indeed, let S : B → B' be an invertible operator and let T: B → B' be another operator. If |S – T | < |S–1|–1, then T is also invertible. This follows by writing T as
and applying the result in the previous section on the second factor. The norm of T–1 can be bounded by
Read more about this topic: Neumann Series
Famous quotes containing the words the set, set and/or open:
“Two principles, according to the Settembrinian cosmogony, were in perpetual conflict for possession of the world: force and justice, tyranny and freedom, superstition and knowledge; the law of permanence and the law of change, of ceaseless fermentation issuing in progress. One might call the first the Asiatic, the second the European principle.”
—Thomas Mann (1875–1955)
“Please do not take counsel of women who are so prejudiced that, as I once heard said, they would not allow a male grasshopper to chirp on their lawn; but out of your own great heart, refuse to set an example to such folly.”
—Frances E. Willard (1839–1898)
“One merit in Carlyle, let the subject be what it may, is the freedom of prospect he allows, the entire absence of cant and dogma. He removes many cartloads of rubbish, and leaves open a broad highway. His writings are all unfenced on the side of the future and the possible. Though he does but inadvertently direct our eyes to the open heavens, nevertheless he lets us wander broadly underneath, and shows them to us reflected in innumerable pools and lakes.”
—Henry David Thoreau (1817–1862)