I am a little bit confused after seeing the definition of inner product and hermitian inner product. Are there any difference in between them. The axioms look like the same in both the cases. In hermitian we have this. Can anyone especially point me out the difference if any? I didn't find this kind of question on google too.
2026-04-02 08:33:05.1775118785
Are there any difference in between inner product and hermitian inner product?
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Well,Hermitian product are antilinear in the second component. This is a strong and important assumption. Inner products in complex spaces do not really make sense,because we can't say whether they are positive or not. While Hermitian forms give us to chance to put a natural norm on these spaces. Clearly,the non linearity in both components do not always behave well:take Rietzs-Fischer theorem for finite dimensional space. If one has a real vector space with a real inner product positive definite ,one gets the canonical identification $V \cong V*$. This clearly does not work for Hermitian forms(the function defined between these spaces would not be linear)!