Undergrad Proof of 'Any Finite Dimensional Unitary Space is Complete'?

[kent davidge]

In texts treating Hilbert spaces, it's usually given as an example that "any finite dimensional unitary space is complete", but I've found no proof so far and failed prove it myself.

 

[fresh_42]

In texts treating Hilbert spaces, it's usually given as an example that "any finite dimensional unitary space is complete", but I've found no proof so far and failed prove it myself.

Unitary means we have a complex (real) vector space with a complex (sesquilinear / real: bilinear) inner product, which is also called a pre-Hilbert space or inner product space. If it is of finite dimension, then we the inner product defines an isometric isomorphism on $\mathbb{C}^n$ (resp. $\mathbb{R}^n\;$) which is complete.