Consider the set of all Kronecker delta functions in some finite dimensional vector space. For instance, in $R^4$, you get the set $\{(1,0,0,0), (0,1,0,0), (0,0,1,0), (0,0,0,1)\}$, which is a simplex.
Informal statement of the question: what if you do this in an $Lp$ space on the interval $[0,1]$?
Obviously this isn't the right way to phrase it, because a Kronecker delta function is equivalent to the zero vector in an $Lp$ space, as it has measure zero. But you could work in a space of distributions instead, induce the $Lp$ norm on that, and then use the set of all Dirac delta distributions on the interval $[0,1]$, yielding something like an "uncountable simplex."
Taken as a manifold, is this topologically or geometrically equivalent to anything more familiar? I thought it might be a hypersphere but that obviously can't be right, as every point is the same distance from every other.
You can also construct this as a direct limit, if you like.