Each vector space $E$ over $\mathbb{C}$ is also a vector space over $\mathbb{R}$. Show that if $(E, \langle \cdot , \cdot \rangle)$ is a complex inner product space, then
$\langle f,g \rangle_r :=$ Re$\langle f,g \rangle \quad f,g \in E$
is a real inner product on $E$ satisfying $\langle if, ig \rangle_r = \langle f,g \rangle_r$ for all $f,g \in E$.
Conversely, show that if $\langle \cdot , \cdot \rangle$ is a real inner product on the $\mathbb{C}$-vector space $E$ such that $\langle if,ig \rangle = \langle f,g \rangle$ for all $f,g \in E$, then
$\langle f,g \rangle_c := \langle f,g \rangle + i \langle f, ig \rangle$
is the unique complex inner product on $E$ with $\langle \cdot , \cdot \rangle_{cr} = \langle \cdot, \cdot \rangle$.
Based on what I know, then I would have to show that the inner product $\langle f,g \rangle_r$ satisfies the four properties in my textbook: sesquilinearity, symmetric, positive, and definite. Is this the right direction to go in order to solve this problem?
The textbook that I am using is Functional Analysis An Elementary Introduction by Markus Hasse. Any help on this problem will be greatly appreciated since I am unsure whether this is the appropriate way to handle it. Thanks in advance.
Yes, except when the product is real then it's not sesquilinear it's just linear in the first argument.
So for the first part you need to show:
(1) $\langle f,g \rangle_r = \langle g,f \rangle_r$
(2) For $\lambda \in \mathbb R$ we have $\lambda \langle f,g\rangle_r = \langle \lambda f,g \rangle_r$ and for $h \in E$ we have $\langle f+h,g\rangle_r = \langle f,g \rangle_r + \langle h,g\rangle_r$
(3) $\langle f,f \rangle_r \ge 0$
(4) $\langle f,f \rangle_r = 0$ if and only if $f = 0$
and in addition to this you need to show the condition $\langle if,ig \rangle_r = \langle f,g\rangle_r$.
For the second part you need to show that $\langle f,g \rangle_c$ is a complex inner product:
(1) $\langle f,g \rangle_c = \overline{\langle g,f \rangle_c}$
(2) For $\lambda \in \mathbb C$ we have $\lambda \langle f,g\rangle_c = \langle \lambda f,g \rangle_c$ and for $h \in E$ we have $\langle f+h,g\rangle_c = \langle f,g \rangle_c + \langle h,g\rangle_c$
(3) $\langle f,f \rangle_c \ge 0$
(4) $\langle f,f \rangle_c = 0$ if and only if $f = 0$
I'm a bit confused about your remaining condition since you never define $\langle f,g\rangle_{cr}$. I presume you want to show that the real part of $\langle f,g\rangle_{c}$ equals $\langle f,g \rangle$ but I might be wrong.