Lp Space: an intuitive understanding. Would someone be willing to give me an intuitive understanding of the Lp space? I have several analysis books which (seemingly) approach the topic differently, which confuses me more, and a simple Google search hasn't led me to any greater enlightenment either. I would be grateful for any feedback from the community. 
Thanks in advance.
 A: I like to look at $L_p$ spaces like generalizations of Euclidean spaces. You can imagine them like Euclidean spaces, but with vectors having infinite number of coordinates. Here the "vectors" are not finite, like ($x_1$, $x_3$, $x_3$, ... $x_n$), but infinite. In the discrete case (small $l_p$ spaces), which is simpler to visualize, the "vectors" have "coordinates" ($x_1$, $x_3$, $x_3$, ... $x_n$, $x_{n+1}$, $x_{n+2}$...) - its coordinates form infinite sequences.
In the continuous case ($L_p$ spaces), the "coordinates" are not separate for each n, but their domain is continuous - of course, since $L_p$ are function spaces.
$L_p$ spaces are important for studying properties and interactions of functions that live there.
Just as vector quantities live in vector spaces, functions f: X $\to$ ${\Bbb C}$ live in function spaces $L_p$. This space also provides some strucutre for the world where functions live. It has a structure of vector space, some algebraic structure (we can multiply functions), metric structure (we can tell if two functions in $L_p$ are "close" or "apart"), topological structure (useful for studying various kind of convergence), norm structure (distinguishing "large" functions from "small" ones) etc.
