# Hermitian, complex inner product space

• December 16th 2008, 08:15 AM
ericmik
Hermitian, complex inner product space
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.

thank u!(Happy)
• December 16th 2008, 08:44 AM
Opalg
Hint: polarisation. The polarisation identity tells you that if $\langle Tx,x\rangle = \langle x,Tx\rangle$ for all x (which is equivalent to $\langle Tx,x\rangle$ always being real), then $\langle Tx,y\rangle = \langle x,Ty\rangle$ for all x and y (which is the definition of T being hermitian).