Direct sum of Banach spaces | Wikipedia audio article
Channel: wikipedia tts
Date: 17/05/2019

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Direct_sum_of_modules


00:00:44 1 Construction for vector spaces and abelian groups
00:01:15 1.1 Construction for two vector spaces
00:03:26 1.2 Construction for two abelian groups
00:05:11 2 Construction for an arbitrary family of modules
00:09:24 3 Properties
00:15:08 4 Internal direct sum
00:16:04 5 Universal property
00:17:32 6 Grothendieck group
00:18:26 7 Direct sum of modules with additional structure
00:19:22 7.1 Direct sum of algebras
00:23:21 7.2 Composition algebras
00:23:40 7.3 Direct sum of Banach spaces
00:24:38 7.4 Direct sum of modules with bilinear forms
00:25:32 7.5 Direct sum of Hilbert spaces
00:26:24 8 See also
00:27:15 9 References
00:28:28 Direct sum of Hilbert spaces
00:31:03 N and Xi
00:32:36 See also



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SUMMARY
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In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module. The direct sum of modules is the smallest module which contains the given modules as submodules with no "unnecessary" constraints, making it an example of a coproduct. Contrast with the direct product, which is the dual notion.
The most familiar examples of this construction occur when considering vector spaces (modules over a field) and abelian groups (modules over the ring Z of integers). The construction may also be extended to cover Banach spaces and Hilbert spaces.