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# Descriptive complexity of Hamel bases of R^w

(base theory = ZFC)

Are any Hamel bases for the vector space $\mathbb{R}^{\omega}$ in the

1. analytical hierarchy?
2. projective hierarchy?

In any of the above cases where the answer is not simply "no", is anything known about what levels they are or can be in?

My knowledge of descriptive set theory is basically just what's on wikipedia, so I probably won't know other theorems even if they are proved in every textbook on the subject. However, I suspect the answers will be
"1. no; 2. none are below $\Delta^1_n$, if V=L then they are in $\Delta^1_n$, if projective determinacy then no"
with n a small natural number explicitly known but not to me.