Dual vector space | Wikipedia audio article
Channel: wikipedia tts
Date: 28/12/2018

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Dual_space


00:01:18 1 Algebraic dual space
00:02:53 1.1 Finite-dimensional case
00:03:05 1.2 Infinite-dimensional case
00:05:43 1.3 Bilinear products and dual spaces
00:06:04 1.4 Injection into the double-dual
00:06:18 1.5 Transpose of a linear map
00:08:31 1.6 Quotient spaces and annihilators
00:11:16 2 Continuous dual space
00:15:15 2.1 Examples
00:19:30 2.2 Transpose of a continuous linear map
00:22:54 2.3 Annihilators
00:24:47 2.4 Further properties
00:26:16 2.5 Topologies on the dual
00:26:41 2.6 Double dual
00:27:11 3 See also
00:31:21 4 Notes
00:43:15 5 References
00:43:44 1.  Then the continuous dual of ℓ p is naturally identified with ℓ q: given an element φ ∈ (ℓ p)′, the corresponding element of ℓ q is the sequence (φ(en)) where en denotes the sequence whose n-th term is 1 and all others are zero. Conversely, given an element a
00:44:17 ∑n anbn for all b
00:45:35 Transpose of a continuous linear map
00:48:44 V and X
00:49:23 Annihilators
00:51:09 Further properties
00:51:34 Topologies on the dual
00:51:55 Double dual
00:55:04 See also



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SUMMARY
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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 the vector space structure of pointwise addition and scalar multiplication by constants.
The dual space as defined above is defined for all vector spaces, and to avoid ambiguity may also be called the algebraic dual space.  When defined for a topological vector space, there is a subspace of the dual space, corresponding to continuous linear functionals, called the continuous dual space.
Dual vector spaces find application in many branches of mathematics that use vector spaces, such as in tensor analysis with finite-dimensional vector spaces. 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.