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In functional analysis, the concept of an orthonormal basis can be generalized to arbitrary (infinite-dimensional) inner product spaces.

this is a exemple on My book ( i am supposed to find a singular value decomposition) well My question is in the book when they use gram-Schmidt to extand they use $ \displaystyle (u_1,u_2,e_3)$ but I would use $ \displaystyle (u_1,u_2,e_1)$ cause it is orthogonal against u_2 which make the gram-Schmidt easy! Edit: I am pretty sure it works cause it is orthonormal to the other! Regards, $ \displaystyle |\pi\rangle$ When you are asked to derive an orthonormal set from a set of vectors, different orders will give different results but they will all be orthonormal sets. (Unless the order is specifically mentioned in the problem.) When you are asked to derive an orthonormal set from a set of vectors, different orders will give different results but they will all be orthonormal sets. (Unless the order is specifically mentioned in the problem.) Petrus, what you say is true, but in the case of a standard basis vector we have: $\text_(\mathbf_j) = \dfrac \mathbf$ If $\mathbf$ is already a unit vector, this becomes: $v_j\mathbf$.In this case, the orthonormal basis is sometimes called a Hilbert basis for H.Note that an orthonormal basis in this sense is not generally a Hamel basis, since infinite linear combinations are required.So if we pick $\mathbf_3 = \mathbf_3$ (as your text does), then Gram-Schmidt gives: $\mathbf_3 = \mathbf_3 - \text_(\mathbf_3) - \text_(\mathbf_3)$ $= (0,0,1) -\dfrac\left(\dfrac, \dfrac,\dfrac\right) - \dfrac\left(0,\dfrac,\dfrac\right)$ $= (0,0,1) - \left(\dfrac,\dfrac,\dfrac\right) - \left(0,\dfrac,\dfrac\right)$ $= \left(\dfrac,\dfrac,\dfrac\right)$ which upon normalization clearly becomes the $\mathbf_3$ in your text.Yes, you are correct that if we pick $\mathbf_3 = \mathbf_1$, then one of the projection terms we subtract is 0, and we get: $\mathbf_3 = (1,0,0) - \dfrac\left(\dfrac, \dfrac,\dfrac\right)$ $= (1,0,0) - \left(\dfrac,\dfrac,\dfrac\right)$ $= \left(\dfrac,\dfrac,\dfrac\right)$ This is the negative (upon normalization) of the vector your book found, and is clearly also perpendicular to the plane spanned by $$.