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Let $V$ be a finite dimensional vector space over $\mathbb{R}$ and $\left<~,\right>$ be a scalar product on $V$.

Is there any obvious extension of $\left<~,\right>$ to $V^*$ the dual of $V$.

Is there any obvious extension of $\left<~,\right>$ to $\Lambda^k V$ the $k$ th exterior power of $V$.

I was thinking about the assignment $\left=\left$ where $u\in V$ such that $f(x)=\left$ for all $x\in V$ and $g(x)=\left$ for all $x\in V$ which exists by Riesz Representation theorem. I am not quite sure if this is what the author is referring to.

For $w_1\wedge w_2\cdots\wedge w_k,v_1\wedge v_2\cdots\wedge v_k$ define $\left=\sum_{i,j}(-1)^{i+j}\left$. I am not quite sure if this is what the author is referring to.

This came when I was reading Complex Geometry - An Introduction by Daniel Huybrechts.

Any suggestion is welcome

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Your suggestion for $V^{*}$ is indeed a natural one and often used. A metric on $V$ gives you an isomorphism $T\colon V \rightarrow V^{*}$ (using Riesz Representation Theorem) and then you put an inner product on $V^{*}$ which turns $\phi$ into an isometry (so you define $\left< \varphi, \psi\right>_{V^{*}} := \left< T^{-1}(\varphi), T^{-1}(\psi) \right>_V$). Note that this means that if $(e_1,\dots,e_n)$ is an orthonormal basis then $(e^1 = T(e_1), \dots, e^n = T(e_n))$ is the dual basis of $(e_1,\dots,e_n)$ and so we see that the inner product on $V^{*}$ makes $e^1,\dots,e^n$ into an orthonormal basis of $V^{*}$.

Regarding the exterior powers $\Lambda^k(V)$, a natural choice is to define an inner product on decomposable $k$-wedges by the formula

$$ \left< v_1 \wedge \dots \wedge v_k, w_1 \wedge \dots \wedge w_k \right>_{\Lambda^k(V)} := \det \left( \left< v_i, w_j \right> \right)_{i,j=1}^k $$

and then to extend using bilinearity to all $\Lambda^k(V)$. The reason this is natural is that if $(e_1,\dots,e_n)$ is an orthonormal basis for $V$, then $(e_{i_1} \wedge \dots \wedge e_{i_k})_{I = (i_1 < \dots < i_k)}$ is an orthonormal basis for $\Lambda^k(V)$.

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    Can you give some reference to support your idea – 2017-02-15
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    See exercise 13, page 79 in "Foundations of Differential Geometry" by Warner or any text that discusses the Hodge star (for example, http://math.stanford.edu/~conrad/diffgeomPage/handouts/star.pdf). – 2017-02-15
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    I am interested in Hodge star operator and this is useful for me... Can you suggest some reference that help to understand that complex geometry book better.. – 2017-02-15