Describe explicitly all inner products on \(\RR^1\) and on \(C^1\).
Solution
All inner products are obtained from a single positive real constant \(c\): \[ \boxed{(x\mid y)=cxy\quad\text{on }\mathbb R,} \] and, with Hoffman–Kunze’s convention of linearity in the first variable, \[ \boxed{(z\mid w)=c z\overline w\quad\text{on }\mathbb C,} \] where in both cases \(c>0\).
1. Real one-dimensional case.
Let \(e=1\) and put \(c=(e\mid e)\). Positive definiteness gives \(c>0\). Since \(x=xe\) and \(y=ye\), bilinearity gives \[ (x\mid y)=xy(e\mid e)=cxy. \] Conversely, for any \(c>0\), this formula is bilinear, symmetric, and satisfies \((x\mid x)=cx^2>0\) for \(x\ne0\), so it is an inner product.
2. Complex one-dimensional case.
Again set \(e=1\) and \(c=(e\mid e)>0\). For \(z,w\in\mathbb C\), linearity in the first variable and conjugate linearity in the second give \[ (z\mid w)=(ze\mid we)=z\overline w(e\mid e)=cz\overline w. \] Conversely this formula has the required sesquilinearity and conjugate symmetry, and \[ (z\mid z)=c|z|^2>0 \] for \(z\ne0\). Hence these are exactly all inner products on \(\mathbb C^1\) [@HK71].