Symmetrizing and Anti-Symmetrizing Tensors Given any Tensor, we can obtain a symmetric tensor through symmetrising operator.
by
$T_{uv} \rightarrow T_{(uv)}=\frac{1}{n!}(T_{uv}+T_{vu})$ where $n$ is the order of the tensor, and you have to take up all the permutations of the indices possible.
And for anti-symmterising tensors, you have to take up alternating sums of the per mutated tensor components, and you get an anti-symmteric tensor.
Now so this is a map from a tensor space to itself.
Is this map only a change of basis map ? Is this a one-one map ?

Not totally sure about your notation ("$uv$" represents an $n$-tuple and the sum represents summing over all permutations?), but viewed as operators on $\bigotimes^{n}(V^{*})$, the space of all $n$-multilinear functions on $V$, neither symmetrization nor anti-symmetrization is one-one or onto.
For $n > 2$, the images of these maps do not even span $\bigotimes^{n}(V^{*})$, i.e., not every $n$-tensor is the sum of a symmetric and anti-symmetric tensor.
