## Inner product on $\mathbb{R}^n$: Is it P:ℝnℝn→ℝ or is it P:ℝnℝn→ℝ+ ?

I am confused about the definition of the inner product
-is it : $P:\mathbb{R}^n\times \mathbb{R}^n\rightarrow \mathbb{R}$
-or: $P:\mathbb{R}^n\times \mathbb{R}^n\rightarrow \mathbb{R} ^+$
I know that it's $\mathbb{R} ^+$ from my teacher
why it's $\mathbb{R} ^+$ ?
An inner product which is also a dot product is negative sometimes if the angle is between pi/2 and 3pi/2