a vector space on which a norm is defined that is complete.
A complete normed vector space. Metric is induced by the norm: d(x,y) = ||x-y||. Completeness means that every Cauchy sequence converges to an element of the space. All finite-dimensional real and complex normed vector spaces are complete and thus are Banach spaces.
Using absolute value for the norm, the real numbers are a Banach space whereas the rationals are not. This is because there are sequences of rationals that converges to irrationals.
Several theorems hold only in Banach spaces, e.g. the Banach inverse mapping theorem. All finite-dimensional real and complex vector spaces are Banach spaces. Hilbert spaces, spaces of integrable functions, and spaces of absolutely convergent series are examples of infinite-dimensional Banach spaces. Applications include wavelets, signal processing, and radar.
[Robert E. Megginson, “An Introduction to Banach Space Theory”, Graduate Texts in Mathematics, 183, Springer Verlag, September 1998].
- Banach-tarski paradox
banach-tarski paradox mathematics It is possible to cut a solid ball into finitely many pieces (actually about half a dozen), and then put the pieces together again to get two solid balls, each the same size as the original. This paradox is a consequence of the Axiom of Choice. (1995-03-29)
noun a tree of the genus Virola, of Central America: family Myristicaceae the timber of this tree, used esp in Honduras for turning and construction Historical Examples (Shoshoni and banak I.) “From the custom of piercing the noses for the reception of ornaments.” Sign Language Among North American Indians Compared With That Among Other Peoples […]
devoid of freshness or originality; hackneyed; trite: a banal and sophomoric treatment of courage on the frontier. adjective lacking force or originality; trite; commonplace adjective pertaining to a lord or ruler (banat) in Hungary, Croatia, and thereabouts Word Origin Serbo-Croatian ban ‘lord, ruler’ adj. “trite, commonplace,” 1840, from French banal, “belonging to a manor, common, […]
to render or make banal; trivialize: Television has often been accused of banalizing even the most serious subjects.