Confusion between orthogonal sum and orthogonal direct sum

  • #1
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For 2 vector spaces an orthogonal direct sum is a cartesian product of the spaces (with some other stuff) (http://planetmath.org/encyclopedia/OrthogonalSum.html [Broken]), and this orthogonal direct sum uses the symbol, [itex]\oplus[/itex].

However, there's an orthogonal decomposition theorem (http://planetmath.org/encyclopedia/OrthogonalDecompositionTheorem.html [Broken]), which says a vector space can be written as the direct sum of a subspace and it's orthogonal complement. But this direct sum, also denoted by the symbol [itex]\oplus[/itex], is not the cartesian product of the 2 spaces.

So [itex]A\oplus B[/itex], sometimes is the Cartesian product of A and B (the first paragraph), and other times it's [itex]\{a+b|a\in A, b \in B\}[/itex] (the second paragraph).

I have a book which mentions a space X being decomposed into an "orthogonal sum of subspaces", and writes: [itex]X=\oplus_{i=1}^\infty Y_i[/itex]. I think the [itex]\oplus[/itex] here means the definition in the second paragraph is this correct?

Can someone check that my understanding of the difference between an orthogonal direct sum and an orthogonal sum, and my use of terminology in this post is correct.
 
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Answers and Replies

  • #2
morphism
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I think what's confusing you is the difference between an "internal direct sum" (in which you write a vector space as a direct sum of subspaces) and an "external direct sum" (in which you take two vector spaces and add them together, without reference to them being contained in a larger vector space).

Orthogonal decomposition should (in my opinion) always be reserved for the situation where there is an inner product (or something similar) available, so that we can actually talk about things being orthogonal.

Anyway, note that if V is an inner product space and U is a subspace of V then the "orthogonal decomposition" ##V=U\oplus U^\perp## is in fact an internal direct sum.

To answer your question about the terminology in your book, it would help if you told us what the book is, or provided us with more context.
 
  • #3
Fredrik
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If an inner product space X is an internal orthogonal direct sum of two of its subspaces Y and Z, then X is isomorphic to the external orthogonal direct sum of Y and Z.

The former statement means that X has subspaces Y and Z such that Y is orthogonal to Z, and for each x in X, there's a unique pair (y,z) such that y is in Y, z is in Z, and x=y+z. The external orthogonal direct sum of two inner product spaces Y and Z is defined as the the set of ordered pairs (y,z) such that y is in Y and z is in Z, with addition and scalar multiplication defined in the obvious ways, and the inner product defined by ##\langle (y_1,z_1),(y_2,z_2)\rangle=\langle y_1,y_2\rangle_Y+\langle z_1,z_2\rangle_Z##.

If you now define Y' as the subspace that consists of ordered pairs (y,0), and define Z' similarly, then the external orthogonal direct sum of Y and Z is an internal orthogonal direct sum of Y' and Z'.

I think the things I have mentioned in here are the reasons why authors sometimes don't even bother to mention if what they have in mind is an internal orthogonal direct sum or an external orthogonal direct sum.
 
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