What is the multiplication "$\times$" for the ordinary vector space $(\Bbb R^n, +, \times)$ If I recall correctly, a field $F$ is a set $V$ associated with two operators $+,\times$ st. $(V,+)$,$(V,\times)$ are both Abelian groups and in addition it satisfies distributivity of $\times$ over $+$, i.e. $\forall a,b,c\in V,a\times(b+c)=a\times b + a\times c$. A vector space over $F$ adds additional interaction with real number field $(\Bbb R, +,\times)$ where $+,×$ are overloaded to denote the ordinary multiplication of real numbers.



I am curious that such an abstract vector space must be a generalization of our ordinary vector space $(\Bbb R^n, +, \times)$ in which each vector consists of real-number components. I am wondering what is the appropriate multiplication here. It is obviously neither cross product nor inner product. Thank you!
You seem to be confusing two separate but related operations: (1) multiplication in the field, and (2) scalar multiplication in the vector space. The standard vector space $\mathbb R^n$ is defined over the field $\mathbb R$. Multiplication in $\mathbb R$ works exactly the way you expect. We can't properly speak of multiplication in $\mathbb R^n$, however. You don't multiply vectors with each other; rather, you multiply them by scalars. So if $\bf x \in \mathbb R^n$ and $c \in \mathbb R$, we can multiply them to get a new vector $c \bf x$. In this case, each co-ordinate of $\bf x$ is simply scaled by $c$.
the dot product will be used here and the symbol
$ \times $ does not mean for cross product as,

$\alpha ,\beta  \to {\mathbb{R}^n}$

$\begin{gathered}
  \alpha  = \left( {{a_1},{a_2}, \ldots ,{a_n}} \right), \\ 
  \beta  = \left( {{b_1},{b_2}, \ldots ,{b_n}} \right) \\ 
\end{gathered} $

then,

$\begin{gathered}
  \alpha \beta  = \left( {{a_1},{a_2}, \ldots ,{a_n}} \right)\left( {{b_1},{b_2}, \ldots ,{b_n}} \right) \\ 
   = \left( {{a_1}{b_1},{a_2}{b_2}, \ldots ,{a_n}{b_n}} \right) \\ 
\end{gathered} $

So, vector multiplication will be used.

Ii may be helpful for you.