Does Pythagorean law holds in Hilbert space or plane?
I know that Pythagorean law holds in Euclidean plane while Hilbert space is a generalisation of Euclidean space.
I know that finite orthonormal set $S$ in a hilbert space $H$, then Pythagorean law holds.
Can someone explain it elaborately whether Pythagorean law holds in Hilbert plane.
Let $S= \{x_1, \ldots, x_n\}$ be a finite orthogonal set in an inner product space. We have $$\left\|\sum_{i=1}^n x_i\right\|^2 = \left\langle \sum_{i=1}^n x_i, \sum_{j=1}^n x_j\right\rangle = \sum_{i=1}^n \sum_{j=1}^n \langle x_i, x_j\rangle = \sum_{i=1}^n \sum_{j=1}^n \|x_i\|^2\delta_{ij} = \sum_{i=1}^n \|x_i\|^2$$ which is the Pythagorean identity.