Explanations
In metric spaces, the completeness is a property of the metric. It is not a property of the topological space itself. If you move on to an equivalent metric (that is a metric which induces the same topology), the completeness can get lost. Regarding two equivalent norms on a normed vector space, however, one of them is complete if and only if the other one is complete. Therefore, in the case of normed vector spaces, the completeness is a property of the norm topology, which does not depend on the specific norm.
Read more about this topic: Banach Space
Famous quotes containing the word explanations:
“Young children constantly invent new explanations to account for complex processes. And since their inventions change from week to week, furnishing the “correct” explanation is not quite so important as conveying a willingness to discuss the subject. Become an “askable parent.””
—Ruth Formanek (20th century)
“In the nineteenth century ... explanations of who and what women were focused primarily on reproductive events—marriage, children, the empty nest, menopause. You could explain what was happening in a woman’s life, it was believed, if you knew where she was in this reproductive cycle.”
—Grace Baruch (20th century)
“We operate exclusively with things that do not exist, with lines, surfaces, bodies, atoms, divisible time spans, divisible spaces—how could explanations be possible at all when we initially turn everything into images, into our images!”
—Friedrich Nietzsche (1844–1900)