What does this set notation mean?

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Discussion Overview

The discussion revolves around the interpretation of the notation \( A \oplus B \) in the context of set theory, specifically regarding its application to vector spaces and subspaces. Participants explore the meaning of this notation and its relevance in mathematical contexts, particularly in quantum theory.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants note that \( A \oplus B \) is not standard notation for sets and suggest it may refer to a direct sum in contexts where a sum is defined, such as vector spaces.
  • One participant confirms that in their context, \( A \) and \( B \) are indeed subspaces of a vector space, which aligns with the proposed interpretation of the notation.
  • Another participant provides a reference to a specific page in a mathematical text to support their understanding of the notation's use in the context of quantum theory.
  • A participant explains that the direct sum of two vector spaces is the smallest subspace containing all vectors from both spaces, and describes a method for constructing a basis for \( A \oplus B \) using bases from \( A \) and \( B \).

Areas of Agreement / Disagreement

Participants generally agree that the notation \( A \oplus B \) can be interpreted as a direct sum in the context of vector spaces, but there is no consensus on its standard use in set theory. Multiple interpretations and contexts are discussed.

Contextual Notes

The discussion highlights the dependence on the specific context in which the notation is used, particularly in relation to vector spaces and quantum theory. There are references to external sources that may provide additional clarity but do not resolve all uncertainties regarding the notation's standardization.

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
 
Last edited by a moderator:
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 [itex]A\oplus B[/itex].
 
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