In this wikipedia article http://en.wikipedia.org/wiki/Tensor_algebra#Construction
We construct $T(V)$ as the direct sum of vector spaces $T^kV$ for $k=0,1,2,…$ $$ T(V)= \bigoplus_{k=0}^\infty T^kV = K\oplus V \oplus (V\otimes V) \oplus (V\otimes V\otimes V) \oplus \cdots$$ The multiplication in $T(V)$ is determined by the canonical isomorphism $T^kV \otimes T^\ell V \to T^{k + \ell}V$ given by the tensor product, which is then extended by linearity to all of $T(V)$.
My question is about the last sentence : "which is then extended by linearity to all of $T(V)$".
What does this sentence mean? i thought that the multiplication given above covers all $T(V)$ and does not need to be extended since any element in $T(V)$ is a tensor product of $k$ elements of $V$ and hence it belongs to $T^kV$!! thank you for your help.