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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.

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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