# Orthonormality preserved when dualing?

Let $$V$$ be an $$n$$ dimensional vector space with orthonormal basis $$\{e_1, \dots, e_n\}$$. Is it true that the dual base on $$V^\ast$$ is also orthonormal?

I think this should be a true statement, but I'm lost in the definitions. If $$\{e_1, \dots, e_n\}$$ is an orthonormal base for $$V$$, then $$\langle e_i, e_j \rangle = \delta_{ij}.$$

Now in order for this to property preserve when we dualize we should have that $$\langle \varepsilon_i, \varepsilon_j \rangle = \delta_{ij}$$ (if $$\{\varepsilon_1, \dots, \varepsilon_n\}$$ is the dual basis) but the issue is that I have no idea what this inner product should be on the dual $$V^\ast$$. If it's the usual dot product on $$V$$ does it immediatly induce something on $$V^\ast$$ or is it possible even to equip the dual space with an inner product?

• So your real question is in the last sentence. Commented Jan 19, 2023 at 23:02
• @PaulFrost The main question is the one entitled, but I need the result of the last sentence for it presumeably. Commented Jan 19, 2023 at 23:08

$$\newcommand\form[1]{\langle#1\rangle}$$I assume that $$\form{\cdot,\cdot}$$ is some kind of symmetric bilinear form. If the form $$\form{\cdot,\cdot}$$ is non-degenerate, i.e. $$(\forall w.\:\form{v,w} = 0) \implies v = 0$$ then the map $$v \mapsto v^\flat$$ defined by $$v^\flat(w) = \form{v, w}$$ is an isomorphism $$V \to V^*$$. So it has an inverse $$\sharp : V^* \to V$$, and we use this to define a form on $$\epsilon,\delta \in V^*$$ via $$\form{\epsilon,\delta} = \form{\epsilon^\sharp,\delta^\sharp}$$.
Now show that $$e_i^\flat = \epsilon_i$$ and the result you want follows.
Define $$\langle \sum\limits_{k=1}^{n}a_k\epsilon_k, \sum\limits_{k=1}^{n}b_k\epsilon_k \rangle=\langle \sum\limits_{k=1}^{n}a_ke_k, \sum\limits_{k=1}^{n}b_ke_k \rangle$$ and verify that this is an inner roduct on $$V^{*}$$ which makes $$(\epsilon_i)$$ orthonormal.
• What is $\delta_k$? @geetha290krm Commented Jan 19, 2023 at 23:24
• @Walker Sorry, I typed $\delta_k$ for $e_k$. Commented Jan 19, 2023 at 23:26