I'm studying sequence spaces $\ell^p=\{(x_j)_{j\in \mathbf{N}}:\sum_{j\in \mathbf{N}}|x_j|^p<\infty\}$ for $1\leq p<\infty$.
This is a vector space (I'm not sure how to prove it is closed under component-wise addition). Indeed it's a normed vector space under the $p$-norm defined by $\lVert (x_j) \rVert := \left( \sum_{j\in \mathbf{N}}|x_j|^p\right)^{1/p}$ (I still need to prove this defines a norm, I just can't show the triangle inequality for it).
But looking ahead of the usual verifications of well-definedness, I want to ask why we study this norm in particular, why we like it on this sequence space and what the intuition is behind its expression.
For context, I am beginning a course on Functional Analysis and so I have seen that this norm makes $\ell^p$ complete, hence a Banach space. Thanks.
I should also note that the structure of the raising to the power $p$ then taking $p$-th root ensures the scalar homogeneity property of a norm, so that kinda makes sense, but what else is going on intuition wise here. How did the first person studying this think this is a natural idea??
I would say that the $L^2$ norm is natural because of Pythagoras, since the space $L^2$ has a scalar product. The $L^1$ norm is pretty natural, just adding up the absolut values. The $L^\infty$ norm, taking the supremum, is also very natural. If you want to study these spaces, it is interesting to see that they lie in this infinite family of spaces, the $L^p$ spaces. They generalise and interpolate between our very usual spaces $L^1,L^2,L^\infty$. They are very interesting because they have good properties. For example the dual space of $L^p$ is $L^q$, where $q$ is the hölder conjugate of $p$ and another good property is, that there is the hölder inequality. and can be used to understand a lot of modern analysis (for example PDEs).