if assume that $X$ is normed spaces, a closed subspace $Y\subset X$ is called $M-$Ideal, If there is a linear projection $P^\*:X^\*\to X^\*$ such that $ker(P)=Y^{\perp}$ and $$||x^\*||=||P(x^\*)||+||x^\*-P(X^\*)||$$


I'm looking an example of mentioned definition but I couldn't give an example in euclidean spaces with different norm.


I would be appreciated if you give an example. 



Looking an example for following definition of M-ideal on normed spaces
I'm not sure what you mean by "euclidean spaces with different norm". Do you just mean a finite-dimensional normed space?


If you just want examples of M-ideals, then the motivating example is: all closed ideals in $C(X)$, where $X$ is compact Hausdorff, are M-ideals; and every M-ideal in $C(X)$ is a closed ideal in the algebraic sense. In general function algebras, some closed ideals are M-ideals but some are not (this is related to the notion of a *peak point* ).


Examples of M-ideals can be found in the book
["M-Ideals in Banach Spaces and Banach Algebras"](http://page.mi.fu-berlin.de/werner99/mideals.html) by Harmand, Werner and Werner.


