Complexification of a real Hilbert space

Question

The following is problem 1.7 in chapter 1 of Conway's A Course in Functional Analysis.

Let $H$ be a Hilbert space over $\Bbb R$ and show that there is a Hilbert space $K$ over $\Bbb C$ and a map $U: H \to K$ such that

a) $U$ is linear

b) $(Uh_1,Uh_2)=(h_1,h_2)$ for all $h_1,h_2 \in H$

c) for any $k\in K$ there are unique $h_1,h_2\in H$ such that $k=Uh_1+iUh_2.$

For this exercise I put K:=$\{ h_1+ih_2 ; h_1,h_2\in H\}$ and define inner product on K such that $(k,k'):= (h_1,h_1')+(h_2,h_2') +i(h_2,h_1')-i(h_1,h_2')$. Now define $U(h)=h$ for all $h \in H$ and extend $U$ linearly. With these, I can not show that $K$ is complete. Please help me.

Could you pinpoint the step where your completeness proof fails? Are you aware that you are operating on $K=H\times H$? — Lutz Lehmann, Dec 24 '13 at 09:08
Just think about how you prove that $\mathbb{C}$ is complete, given that $\mathbb{R}$ is complete. — Prahlad Vaidyanathan, Dec 24 '13 at 11:35

Martin Argerami · Answer 1 · 2018-03-07T17:32:02.737

For completeness you need to show that if a sequence is Cauchy, then its real and imaginary parts are Cauchy; this follows easily from the fact that $\|h_1+ih_2\|^2=\|h_1\|^2+\|h_2\|^2$ (note that you have a typo in your definition of the inner product).

Note also that you cannot define $K$ the way you do: you have no meaning for $i$ times vector, nor for the plus sign. What you have to do is to let $K=H\times H$ and define addition and multiplication by complex scalars appropriately.

Complexification of a real Hilbert space

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