If $T$ is an operator on inner product space, how do we prove that $T$ is invertible iff $T^*$ is invertible?
Can I change the goal to prove $T$ is injective iff $T^*$ is injective?
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2Look this up-http://math.stackexchange.com/questions/410532/invertibility-in-a-finite-dimensional-inner-product-space – The very fluffy Panda Apr 10 '14 at 22:16
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You should probably be assuming that the underlying space is complete. Otherwise, there's a nice example given of a bounded operator that doesn't have an adjoint: http://math.stackexchange.com/questions/29140/adjoint-of-a-linear-transformation-in-an-infinite-dimension-inner-product-space – Disintegrating By Parts Apr 10 '14 at 23:14