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18.906 Problem Set 7 Due Friday, April 6 in class
18.906 Problem Set 7 Due Friday, April 6 in class

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... Another example is that the set of all real numbers, denoted R, with the usual addition and scalar multiplication and the absolute value as the norm is a Banach space. But we do wonder: If S is complete with respect to one norm, is it complete with respect to other norms? First why do we care about ...
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MATH 412: NOTE ON INFINITE-DIMENSIONAL VECTOR SPACES

... zero vector has a unique representation in the basis; equivalently, every vector can be uniquely represented. 1.2. Infinite dimensions,explicit basis: R⊕N . Let R⊕N denote the space of se∞ quences x = (xn )n=1 which have finite support, that is so that xn = 0 from some point onward. This space is ev ...
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Dual space

In mathematics, any vector space V has a corresponding dual vector space (or just dual space for short) consisting of all linear functionals on V together with a naturally induced linear structure. Dual vector spaces for finite-dimensional vector spaces show up in tensor analysis. When applied to vector spaces of functions (which are typically infinite-dimensional), dual spaces are used to describe measures, distributions, and Hilbert spaces. Consequently, the dual space is an important concept in functional analysis.There are two types of dual spaces: the algebraic dual space, and the continuous dual space. The algebraic dual space is defined for all vector spaces. When defined for a topological vector space there is a subspace of this dual space, corresponding to continuous linear functionals, which constitutes a continuous dual space.
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