Inner product on space of functions from $R^n \rightarrow R^m$?

Question

Is the space of functions from $R^n$ to $R^m$ a Hilbert space? It seems pretty obvious that it's a vector space, but I'm not sure how to extend the inner product on functions $R^n \rightarrow R$ to functions in $m$D. Does this space of functions (possibly with some restrictions such a smoothness) admit an inner product? If so, what is that product? If there are more than one, is there a "natural" one that is related to the single-dimensional function case?

Answer 1

Let $\boldsymbol{f}(\boldsymbol{x}) = \left(f_1(x_1,\dots,x_n),\dots, f_m(x_1,\dots,x_n)\right)^T$ and $\boldsymbol{g}(\boldsymbol{x})$ be functions from $\mathbb{R}^n$ to $\mathbb{R}^m$. In my opinion, the following definition of inner product is quite straightforward: $$ \left<\boldsymbol{f}, \boldsymbol{g}\right> = \sum_{i=1}^m \int_{\mathbb{R}^n} f_i(\boldsymbol{x}) g_i(\boldsymbol{x}) \mathrm{d} \boldsymbol{x} = \int_{\mathbb{R}^n} \sum_{i=1}^m f_i(\boldsymbol{x}) g_i(\boldsymbol{x}) \mathrm{d} \boldsymbol{x}. $$ You can check as an exercise that all needed properties are satisfied. Of course, you need some assuptions on $\boldsymbol{f}$ and $\boldsymbol{g}$ hold, i.e. integrability and $\int_{\mathbb{R}^n} \sum_{i=1}^m f_i^2(\boldsymbol{x}) \mathrm{d} \boldsymbol{x} < \infty$ (and same for $\boldsymbol{g}$).