Is there some meaningful difference between: $f(x)=g(x)+c$ and $f(x)=g(x)+c\operatorname{Id}(x)$?
Because my book defines the eigenspace as follows: $$ E_\lambda=\{v\in V\mid (L-\lambda\cdot\operatorname{Id})(v)=0\}=\ker(L-\lambda\cdot\operatorname{Id}). $$ Would there be some subtle difference if I wrote $(L-\lambda)(v)$? Is the problem here that it now looks as if $\lambda$ is a function of $v$? But it is... it's the constant function on $v$.
I was just wondering if the notation I use is wrong.
$g(x) + c\operatorname{Id}(x)$ is exactly the same as $g(x) + cx$ because $\operatorname{Id}(x) = x$ for all $x$ in your domain.
However, $g+c\operatorname{Id}$ is not the same as $g+c$. Technically speaking, $g+c$ (function $+$ scalar) isn't even defined with the usual definitions. But, expanding on what copper.hat says in the comments, it is a common abuse of notation that literally every mathematician will understand (and most probably won't even notice).