Generalization To Topological Vector Spaces
In the special case of two topological vector spaces and, the notion of uniform continuity of a map becomes: for any neighborhood of zero in, there exists a neighborhood of zero in such that implies
For linear transformations, uniform continuity is equivalent to continuity. This fact is frequently used implicitly in functional analysis to extend a linear map off a dense subspace of a Banach space.
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Famous quotes containing the word spaces:
“When I consider the short duration of my life, swallowed up in the eternity before and after, the little space which I fill and even can see, engulfed in the infinite immensity of spaces of which I am ignorant and which know me not, I am frightened and am astonished at being here rather than there. For there is no reason why here rather than there, why now rather than then.”
—Blaise Pascal (1623–1662)