What does it mean to "fix" an orthonormal basis?

Question

In one of my homework problems it hints to "fix an orthonormal basis of $V$". Does this simply mean to come up with a basis for $V$ like $v_1 ... v_n$? If I fix an orthonormal basis, does it matter if $V$ is infinite dimensional?

Answer 1

To slightly expand my comment:

To "fix a basis" means that we choose a particular basis that we're going to be stuck with until we change it.

And it doesn't matter what the dimension of the vector space is, although there are some interesting complications that come up when we have a basis of an infinite dimensional space. See, for example, the concept of a Hamel basis.

Answer 2

It means to choose an orthonormal basis among the infinitely many.

We can define it also for infinite dimensions.

See the related Orthogonal basis for infinite-dimensional vector spaces