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Prove that $T^*$ is injective iff $ImT$ Is dense The title of your question does not really match the actual question (maybe the statement of the current question is used to prove the result in the title?) Is this intended?
Give an example of a linear map $T$ such that $\dim (\operatorname . . . This is completely correct This will give a linear map with the properties you're asked for I think that it is a bit too general to actually be "an example" I think it would be better if you actually pick a concrete basis But that's a personal aesthetic belief, and one would have to be pretty pedantic about it to say that that makes you wrong One objection with a bit more substance is
Example of linear transformation on infinite dimensional vector space I haven't had much experience with infinite dimensional vector spaces, and I was working on a problem that asks to prove that for a finite dimensional vector space $V$, and linear transformation $T:V\to V$, $V=imT + ker T \implies V=imT \bigoplus ker T$