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Mathematical notation is a system of written symbols and conventions for specifying mathematical objects, operations and relations. Its purpose is to make structure visible while keeping expressions unambiguous. An expression needs meanings for its symbols as well as rules for how they combine.

For instance $f\circ g$ denotes composition in the convention $(f\circ g)(x)=f(g(x))$, so the rightmost function acts first. The summation ${\sum }_{k=1}^{n}k$ binds a dummy index and records both endpoints. Familiar symbols can conceal different conventions; stating index ranges, domains and normalization prevents a compact formula from becoming an ambiguous one.

Relabel the two dummy indices and use the respective symmetries:Thus the symmetric tensor and antisymmetric tensor subspaces are orthogonal in the Frobenius inner product.