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Here is one of my doubts I encountered after studying many linear algebra books and texts. The Euclidean space is defined by introducing the so-called "standard" dot (or inner product) product in the form:

[tex] (\boldsymbol{a},\boldsymbol{b}) = \sum \limits_{i} a_i b_i [/tex]

With that one can define the metric and the vector norm, the latter one as:

[tex] || \boldsymbol{a} || = \sqrt{(\boldsymbol{a},\boldsymbol{a})} [/tex]

etc. etc. However, we know that the first formula is valid only when we chose the orthonormal basis. That is the basis consisting of the vectors which are mutually orthogonal and with unit lengths:

[tex] (\boldsymbol{e}_i,\boldsymbol{e}_j) = \delta_{ij} [/tex]

[tex] || \boldsymbol{a}_i || = 1 [/tex]

Thus, to define the orthonormal basis one need to define dot product and norm first. On the other hand, the dot product formula works only in the case of othonormal basis. If we take any other basis this formula will not be valid (in Euclidean space?). Is that correct? The question is then in what order should be define all the terms to be consistent. Or perhaps there are more Euclidean spaces, each of its metric, and the above choice is arbitrary so it resembles the physical space the most? The question is then why it is like this? I would like to avoid the formula:

[tex] (\boldsymbol{a},\boldsymbol{b}) = || \boldsymbol{a} || \cdot || \boldsymbol{b} || \cdot \cos \theta [/tex]

since for this we need the norm and the angle... Closed loop and one of my doubts on how to deal the the topic...

Many thanks for explanations!

Radek