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If $V$,$W$are two inner product spaces and $L:V\to W$ is a linear map with its adjoint $L^\star$, then is there a decomposition of $W$=ker$(L^\star)$ $ \oplus $ im$(L)$ ? (It is easy that the conclusion holds if $V$ and $W$ are finite-dimensional)

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What if $\operatorname{im}{L}$ isn't closed? –  t.b. Apr 4 '12 at 9:57
    
Which duals do you consider? Algebraic or continuous? –  Norbert Apr 4 '12 at 12:13
    
@Norbert algebraic, since we do not assume that $L$ is continuous. –  Hezudao Apr 4 '12 at 13:17
    
@t.b. I see. If the decomposition does exist, then im($L$) should be closed. But what if we assume first that im($L$) is closed? –  Hezudao Apr 4 '12 at 13:42
    
I think, $L^*$ is not well definied, so you should give your definition. –  Norbert Apr 6 '12 at 5:58
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