I know that in $\mathbf{\mathbb{R}}^n$ the definition of the dot (or scalar) product is the following:
$x.y=x^{\mathrm{T}}y$, with ''T" denoting the transpose of the vector x.
How does this definition change when working in infinite space, e.g:
$\int_{\Omega}\nabla u \cdot \nabla v \, d\Omega = 0,\, \forall v\in H^{1}_{0}\left(\Omega\right)$
am I allowed to write:
$\int_{\Omega}\left(\nabla u\right)^{\mathrm{T}}\, \nabla v\,d\Omega = 0,\, \forall v\in H^{1}_{0}\left(\Omega\right)$ ??
Thanks in advance for your answers
I basically found my answer from this question, where they clarified the ambiguity I had with the definition of the inner product. In fact, when working in an infinite dimension space the inner product (a general nomenclature and definitions for different spaces) is the one used and as the dimension becomes finite its definition becomes the dot product (or you can still call it an inner product). So basically, the dot product is the definition of the inner product in a finite dimension space. Me moving from integral 1 to the second integral defined in a discretized space is completely fine as long as I remind the definition of the inner product.
I confirmed this by the comments I've gotten here.
Hope I was clear enough :)