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This question already has an answer here:

For $V_n$ where $x=(x_1, x_2, \ldots, x_n)$ and $y=(y_1,y_2,\ldots,y_n)$, the dot product is defined by $x_1y_1+x_2y_2+ \cdots+x_ny_n$.

In Apostol's calculs vol 2

It says that if $x=(x_1,x_2)$ and $y=(y_1,y_2)$ are any two vectors in $V_2$, define $(x \cdot y)$ by the formula

$(x \cdot y)= 2x_1y_1+x_1y_2+x_2y_1+x_2y_2$

Why is that?

Is there difference between dot and inner product?

I thought that following the dot product, $(x \cdot y)$ should be $x_1y_1+x_2y_2$.

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marked as duplicate by Jeel Shah, Jonas Meyer, user147263, Peter Woolfitt, Daniel W. Farlow Apr 28 '15 at 19:20

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    $\begingroup$ The dot product is another name for an inner product on $\mathbb{R}^n$ and typically refers to the one you mentioned first. Inner products are more general but share many of the same properties with the usual dot product. $\endgroup$ – Cameron Williams Sep 17 '14 at 4:50
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An inner product is defined on any real or complex vector space by having three properties, the details are well listed here: http://en.wikipedia.org/wiki/Inner_product_space The dot product between two real vectors in a real vector space satisfies all of these properties, so it is AN inner product, but there are others. Also note things get funkier when you shift your field to the complex numbers

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