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For some reason my book distinguishes the two names.

If a set is an orthogonal set, doesn't that make it immediately a basis for some subspace $W$ since all the vectors in the orthogonal set are linearly independent anyways? So why do we have two different words for the same thing?

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    @mim: By convention, if we say "basis" without specifying basis for *what*, we mean the basis of the whole space of discourse.2012-02-23

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When you say orthogonal basis you mean that the set is a basis for the whole given space. Every orthogonal set is a basis for some subset of the space, but not necessarily for the whole space.

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    If you have an orthogonal set in ${\bf R}^n$, then it's not a basis for ${\bf R}^n$ if it has fewer than $n$ members, and it's not a basis for ${\bf R}^n$ if it contains the zero vector, and it is a basis for ${\bf R}^n$ if it has $n$ members and doesn't contain the zero vector.2012-02-23
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The reason for the different terms is the same as the reason for the different terms "linearly independent set" and "basis".

Every linearly independent set is a basis for the subspace it spans. But when working in a larger space "basis" means "maximal linearly independent set" (not just spanning a subspace but spanning the whole thing).

An orthogonal set (without the zero vector) is automatically linearly independent. So we have "orthogonal sets" and then maximal ones are "orthogonal bases".

Note: In the end we're essentially just tacking on the adjective "orthogonal". We don't keep the words "linearly independent" in "orthogonal linearly independent set" because they're redundant.

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  1. These two concepts are totally different.
  2. For "orthogonal set"$M$, we only have Bessel's inequality.
  3. But, if $M$ is orthogonal basis, then we get the Parseval's theorem. The key point is the completeness of this set M in your space. For example, in finite dimensional space $\mathcal{R}^3$, $\{i,j\}$ is an orthnormal set, but not an orthonormal basis. A common orthonormal basis is $\{i,j,k\}$.