The theorem says, "If $P$ be a subspace of a finite dimensional Euclidean subspace of $V$, then $V=P\oplus P^\perp$.
My book starts the proof like this:
Let $P\ne \{{\theta}\}$ and let $\{\beta_1,\beta_2,\ldots,\beta_r\}$ be an orthogonal basis of $P$.
And the proof goes on.
Now my question is what ensures that a subspace must have an orthogonal or orthonormal basis? Does a basis necessarily need to be orthogonal or orthonormal?