Let x = (x1, …, xn) and y = (y1, …, yn) denote vectors in ℝn for a fixed n ≥ 2. Which of the following defines an inner product on ℝn?
1
〈x, y〉 = \(\rm\displaystyle\sum_{i, j=1}^n\) xiyj
2
〈x, y〉 = \(\rm\displaystyle\sum_{i, j=1}^n\left(x_i^2+y_j^2\right)\)
3
〈x, y〉 = \(\rm\displaystyle\sum_{j=1}^n\) j3 xjyj
4
〈x, y〉 = \(\rm\displaystyle\sum_{j=1}^n\) xj yn−j+1