A real inner product space is defined in the same way, except that H is a real vector space and the inner product takes real values. Such an inner product will be a bilinear map and ( H , H , ⟨ ⋅ , ⋅ ⟩ ) {\displaystyle (H,H,\langle \cdot ,\cdot \rangle )} will form a dual system.[4]