On a Hilbert space $H$ with an orthonormal basis $(e_j)_{j \ge 1}$, define $E$ to be spanned by $e_1 + e_2, e_3 + e_4, $ and $e_2 + e_3$.
What's an orthonormal basis for $E$?
I have no idea where to begin. Usually we are given some vectors, and then I'd apply Gram-Schmidt, but what now?
Hint: We don't lose any information if we represent these vectors as column vectors. In particular, set $$ e_1 = \pmatrix{1\\0\\0\\0}, \dots, e_4 = \pmatrix{0\\0\\0\\1} $$ Now, apply Gram Schmidt.
Note, however, that we don't necessarily need coordinate vectors to apply Gram-Schmidt. In particular, we begin the the vectors $$ v_1 = e_1 + e_2\\ v_2 = e_3 + e_4\\ v_3 = e_2 + e_3 $$ We want to find an orthonormal basis $u_1,u_2,u_3$ of the span. We begin by saying $$ u_1 = \frac{1}{\|v_1\|}v_1 = \frac 1{\sqrt{\langle e_1 + e_2, e_1 + e_2 \rangle}}(e_1 + e_2) $$ Note, however, that $$ \langle e_1 + e_2, e_1 + e_2 \rangle = \langle e_1, e_1 \rangle + \langle e_2, e_1 \rangle + \langle e_1, e_2 \rangle + \langle e_2, e_2 \rangle =\\ 1 + 0 + 0 + 1 = 2 $$ so, we have $$ v_1 = \frac 1{\sqrt 2} e_1 + \frac 1{\sqrt 2} e_2 $$ perhaps you can take it from there.