High School What does this set notation mean?

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The notation A ⊕ B is not standard for sets but can represent the direct sum of two vector spaces when A and B are subspaces. In this context, the direct sum is defined as the smallest subspace containing all vectors from both A and B. To construct A ⊕ B, one can combine the bases of A and B and reduce them to a set of independent vectors. This clarification aligns with the reference provided from Brian Hall's work on quantum theory. Understanding this notation is crucial for discussing vector space properties effectively.
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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
 
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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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