Let $A=T(V)$ be the tensor algebra and let $\{v_1,...,v_n\}$ be a basis for $V$, then we get a basis for $A$ by taking tensor products of $\{v_1,...,v_n\}$. I read in Koszul Algebras by R. Fröberg that $$Hom(A,\mathbb{k})=\oplus_{i\geq 0}Hom(A_i,\mathbb{k})$$ But is there not a functional $f$ taking all the basis vectors of $A$ to 1? And if so, would $f$ not be an infinite sum $\sum_{i\geq0}f_i$ where $f_i\in A_i$ takes all the basis elements of $A_i$ to 1?
Express the dual space of the tensor algebra as a direct sum
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linear-algebra
direct-sum