While studying inner product space I came across a theorem that is if V is an inner product space and $\{x_1,x_2,.....x_n\}$ be the orthonormal set of nonzero vectors. Then if $y\in V$ then $ y=\sum_{i=1}^{i=n} <y,x_i>x_i$.
Now my doubt is if there are two different inner product space, can $y$ be written as in two different combination of that same element that is $\{x_1,x_2,.....x_n\}$ is orthonormal set of elements for the second inner product space also.
If what I am stating is true, then it will be great if someone can provide me an example proving my claim. Any help will be appreciated. Thanks
If the $x_i$ are orthonormal, then they are linearly independent, so there is at most one way to represent $y$ as a linear combination of them, regardless of what inner product you use.