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Assume I have a sequence of vectors $v_k \in V$ in some abstract vector space $V$ not necessarily equal to $\mathbb{C}^n$. Can I still use the Householder QR decomposition in this case, even though the concept of "reflecting onto the first coordinate direction" doesn't make sense if $V \neq \mathbb{C}^n$? Or do I have to use the less stable modified Gram-Schmidt process to find an orthogonal basis for $\DeclareMathOperator{Span}{span}\Span\{v_k\}$?

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If $V$ is an inner product space, you can define Householder operators that generalize Householder reflectors. If $V$ is finite-dimensional and you have an orthonormal basis with respect to the inner product of $V$, you should be able to compute QR via Householder reflectors. If you only have a basis, you could orthogonalize it via (modified) Gram-Schmidt and then proceed as above.

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