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## Homework Statement

Suppose V is an inner product space and [tex]T\in L(V)[/tex]. Prove that if [tex]\left\|T^{*}v\right\|[/tex][tex]\leq[/tex][tex]\left\|Tv\right\|[/tex] then T is Normal.

Sorry for being bothersome; this is the first year I've ever written proofs so I'm a bit sketchy about them.

## Homework Equations

## The Attempt at a Solution

Define an Inner Product on V [tex]\left\langle Sv, Tv\right\rangle[/tex] = [tex]\ Trace(ST^{*})[/tex].Then [tex]\left\|T^{*}v\right\|[/tex][tex]\leq[/tex][tex]\left\|Tv\right\| = Trace (TT^{*}) \leq Trace( T^{*}T)[/tex].

However, [tex]\ Trace (TT^{*}) = Trace (T^{*}T)[/tex]. Which implies [tex]\left\|T^{*}v\right\|[/tex] = [tex]\left\|Tv\right\|[/tex], and therefore T is normal

I worked up another solution, which I'm almost sure is correct, but if this one does indeed check out I'd prefer to use it ;p. It just seems wrong to me for some reason; because it implies that basically all linear maps are normal; but I can't place what I did incorrectly.

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