Is Minkowski space a Hilbert space?

Question

In other words, is Minkowski space a real or complex inner product space that is also a complete metric space with respect to the distance function induced by the inner product? I would think that Minkowski space can either be thought of as 2-D complex or 4-D real, do either of these interpretations change whether or not it is a Hilbert space?

Answer 1

No. In Minkowski space we have a bilinear form$$\bigl\langle(t,x,y,z),(t',x',y',z')\bigr\rangle=-tt'+xx'+yy'+zz'$$which is not an inner product. Therefore, it doesn't induce a metric.

Answer 2

A Hilbert space is a real or complex vector space with an inner product which induces a norm under which the space is complete. Minkowski space incorporates three dimensional space together with a time dimension. It comes with a Lorentzian inner product which is not an inner product (it is not positive definite). When one says 'Minkowski space' one usually means the geometry of the space with the Lorentzian inner product. Therefore, no, Minkowski space is not a Hilbert space.