Now that we have the Gram-Schmidt process as a tool, we can use it to come up with orthonormal bases.
Any vector space with finite dimension has a finite basis . This is exactly what it means for to have dimension . And now we can apply the Gram-Schmidt process to turn this basis into an orthonormal basis .
We also know that any linearly independent set can be expanded to a basis. In fact, we can also extend any orthonormal collection of vectors to an orthonormal basis. Indeed, if is an orthonormal collection, we can add the vectors to fill out a basis. Then when we apply the Gram-Schmidt process to this basis it will start with , which is already normalized. It then moves on to , which is orthonormal with , and so on. Each of the is left unchanged, and the are modified to make them orthonormal with the existing collection.