We know that if $X$ is an inner product space with a base $B$, then we can construct (by Gram-Schmidt) an orthogonal base $B'$ for $B$.
But, if we only have a linearly independent subset of $X$, lets say it $A=\{x_n\}_{n \in \mathbb{N}} $ then can we construct a orthogonal set $A'$ from $A$ by doing the same ( Gram-Schmidt) process ?.
And if so, will they have the same $Span$?
Best Answer
Yes. Note that the Gram-Schmidt process requires only a linearly independent set (and all bases are linearly independent).
A proof that this works is by induction. Suppose that $\{x_k\}_{1 \leqq k \leqq n}$ is a linearly independent set of vectors, then (note: we will normalise the vectors at the end because this substantially simplifies things)