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I understand that, for example the inner product space $L^2(X)$ of complex - valued functions defined on $X$ has the inner product \begin{equation} (f,g) = \int f \, \overline{g\,}. \end{equation}

Now I am reading a text where the elements of my inner product space are vector - valued functions, and it is assumed the reader knows how to adjust the inner product so that it works in this case. I am not sure how to do this, here is a guess and it would be great if I could get feedback on whether this is the right way to generalize to vector - valued functions:

\begin{equation} (f,g) = \sum_{i = 1}^n \int f_i \, \overline{g_i} \end{equation}

(here, $n$ is the dimension of the range, and $f_i$ is the $i^{th}$ component of the function $f$). Many thanks !

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Check if your definition fits axioms of the scalar product and does it give you a desired norm. – Ilya Apr 22 '12 at 11:16
@Ilya: thanks for your suggestion, if I have done eveerything right then it does give a scalar product. In order to be sure that's the the generalization I would also have to show that this is the unique way to generalize the inner product given above - is that right ? – harlekin Apr 22 '12 at 11:26
@harlekin You are correct in that it is indeed an inner product. There is no "right" inner product, an inner product is an inner product. However, your guess is indeed the most usual and canonical inner product on that particular space and is most likely the one your text was referring to. – Ragib Zaman Apr 22 '12 at 11:48
up vote 4 down vote accepted

If you work with vector values functions, say in a inner product space $E$, you can define $$(f,g):=\int_X \langle f(x),g(x)\rangle d\mu,$$ where $\langle \cdot,\cdot\rangle$ is an inner-product on $E$. The properties of integration and inner-product ensure us that $(\cdot,\cdot)$ is an inner-product over $L^2(X,E)$.

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