I am a bit confused regarding the correct formula for the complex inner product of two complex vectors. The textbook I learnt from defines the complex inner product of the vectors: $u$ and $v$ as follows: $\left ( cu,v \right )=\bar{c}\left ( u,v \right )$ and $\left ( u,cv \right )=c\left ( u,v \right )$ where $c$ is a scalar.
However, in a different textbook, I found the following convention: $\left ( cu,v \right )=c\left ( u,v \right )$ and $\left ( u,cv \right )=\bar{c}\left ( u,v \right )$
Which one of the above formulas is the one I should use?
Also, for the complex dot product: In some textbooks, I found: $u\cdot v=\bar{u_{1}}v_{1}+\bar{u_{2}}v_{2}+\cdots+\bar{u_{n}}v_{n}.$ In other textbooks, I found the following: $u\cdot v= u_{1}\bar{v_{1}}+u_{2}\bar{v_{2}}+\cdots+u_{n}\bar{v_{n}}$
- Which one I should as the definition of the complex dot product? Also, which one is equivalent to $u^{*}v$?