Hint: polarisation. The polarisation identity tells you that if for all x (which is equivalent to always being real), then for all x and y (which is the definition of T being hermitian).
i'd be happy with any assistance.
Let T be a linear operator on a complex inner product space V .
Prove that if T is Hermitian, then ⟨T (x), x⟩ is real for all x ∈ V .
also, Prove theat if ⟨T (x), x⟩ is real for all x ∈ V , then T is Hermitian.