Usually, a $k$-simplex is defined as a subset of $\mathbb{R}^n$ spanned by $k+1$ vertices $v_0,\ldots , v_{k}$, that is the set of linear combinations of the vertices with real non negative coefficients whose sum is $1$. I have two questions: is it possible to substitute $\mathbb{R}^n$ with any real vector space? Has that vector space to be necessarily finite dimensional over $\mathbb{R}$?
2026-04-03 14:31:48.1775226708
On the definition of k-simplex
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Nothing in the definition of simplex stops you from defining a $k$-simplplex in an infinite dimensional vector space, so the answer to your second question is 'no the vector space does not need to be finite-dimensional'.
As for the first question, as long as you are able to form finite linear combinations you can form a simplex, albeit if you find yourself dealing with a vector space without an inner product, you may have a hard time verifying whether or not the $v_0,\ldots,v_k$ are in general position.