What does this set notation mean?

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SUMMARY

The notation $$A \oplus B$$ refers to the direct sum of two sets A and B, specifically when these sets are subspaces of a vector space. This notation is not standard for general sets but is applicable in the context of vector spaces, as discussed on page 137 of Brian Hall's "Quantum Theory for Mathematicians" (2013). The direct sum represents the smallest subspace containing all vectors from both A and B, achieved by constructing bases for each and reducing them to independent vectors.

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pellman
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$$
A \oplus B
$$

where A and B are sets
 
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That is not a standard notation for sets. It can mean a direct sum for sets with sum kind of "sum" defined, such as vector spaces. Is that what you mean?
 
HallsofIvy said:
That is not a standard notation for sets. It can mean a direct sum for sets with sum kind of "sum" defined, such as vector spaces. Is that what you mean?
Actually, yes, in the context it was used the sets in question are subspaces of a vector space.

The context is page 137 here http://perso.crans.org/lecomtev/articles/Brian_Hall_Quantum_Theory_for_Mathematicians_2013.pdf
 
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pellman said:
Actually, yes, in the context it was used the sets in question are subspaces of a vector space.

The context is page 137 here http://perso.crans.org/lecomtev/articles/Brian_Hall_Quantum_Theory_for_Mathematicians_2013.pdf

https://en.wikipedia.org/wiki/Hilbert_space#Direct_sums
 
Last edited by a moderator:
The "direct sum" of two vector spaces, A and B, (both subspaces of some vector space, V) is the smallest subspace that contains all the vectors in both A and B. Another way of doing that is to construct bases for both A and B, combining them and then reducing to a set of independent vectors to get a basis for A\oplus B.
 
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