Tensor topologies on spaces of symmetric tensor products

Here we show how, in the general context of locally convex spaces, it is possible to get an n-tensor topology (on spaces of n-tensor products) from an n-tensor topology on spaces of symmetric n-tensors products. Indeed, given an n-tensor topology on the spaces of symmetric n-tensor products we construct an n-tensor topology on the spaces of all n-tensor products whose restriction to the symmetric ones gives the original topology. Moreover, we prove that when one starts with an n-tensor topology, restricts it to symmetric tensors and then extends it, the original topology is obtained when it is symmetric, and we also obtain some results on complementation with applications to spaces of polynomials. Part of these results generalize to the context of locally convex spaces some Floret's results in [17] and [18].