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Math Help - Hermitian, complex inner product space

  1. #1
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    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!
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  2. #2
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    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).
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