  We prove that the period doubling operator has an expanding direction at the
fixed point. We use the induced operator, a ``Perron-Frobenius type operator'',
to study the linearization of the period doubling operator at its fixed point.
We then use a sequence of linear operators with finite ranks to study this
induced operator. The proof is constructive. One can calculate the expanding
direction and the rate of expansion of the period doubling operator at the
fixed point.
