I'm almost sure that I'm misreading the theorem, but I have no idea where.
Theorem 1.13 is about orthogonal bases and Gram-Schmidt process. It states:
Let $x_1, x_2, ...., $ be finite or infinite sequence of elements in a Euclidean space V and let $L(x_1, ...., x_k)$ denote the subspace spanned by the first $k$ of these elements. Then there is a corresponding sequence $y_1, y_2, ...,$ in V which has the following properties for each integer $k$:
a) the element $y_k$ is orthogonal to every element in the subspace $L(y_1, y_2, ...y_{k-1})$
b) the subspace spanned by $y_1, ...,y_k$ is the same as that spanned by $x_1, ...,x_k$
$L(y_1,...,y_k)=L(x_1,...x_k)$
(c) the sequence $y1, y2, ....$ is unique except for scalar factors. That is if $y_1',y_2'...$ is another sequence of elements in V satisfying properties (a) and (b) for all $k$, then for each $k$ there is scalar $c_k$ such that $y_k'=c_ky_k$.
(c) seems to me obviously wrong.
in $R^2$ we can have u and v span the whole space. We can have w and a span the space too. But I think we cannot produce either a either w by scalar multiplication of u or v.
We can take any orthogonal base and rotate it in space to get another one. That new one cannot be the "same" up to multiplication of the individual vectors by scalars. Right???
Is it possible that he refers to the base that is produced by Gram-Schmidt?
The point is that such $(y_k)$ sequence is unique for each given $(x_k)$ sequence. Then yes, there are infinitely many orthogonal sequences that span the same space, but if you are given an initial sequence $(x_k)$, the corresponding $(y_k)$ will be unique except for multiplication by scalars.
To visualize that, pick $y_1 = x_1$, since $y_1$ must span the same space as $x_1$ (by assertion $b$). Then we need to have $y_2$ being perpendicular to the space generated by $y_1$ and at the same time ${y_1, y_2}$ generating the same space as ${x_1,x_2}$, therefore it is necessary to have $y_2$ being a linear combination of $x_1$ and $x_2$ perpendicular to $x_1$ (and consequently $y_1$). The theorem result will follow inductively with this reazoning.