Let $X$ be an inner product space, $M \subset X$ and $Y \subset M^{\perp} $.
Let each $u \in X$ has a representation $u = v +w$ with $v \in M$ and $w \in Y.$
So I need to show that : $Y = M^{\perp}$ and to do so, I only need to prove that $M^{\perp} \subset Y$.
For each $u \in X$, we also have $u = v^{\sim}+w^{\sim}$ where $V^{\sim} \in M$ and $s^{\sim} \in M^{\perp}.$ Thus equating the two representations of $u$, one can write $u+v= v^{\sim}+w^{\sim}$ All I am thinking so far is to show that $u = u^{\sim}$ and $v=v^{\sim}$. but I am not sure if it the right approach and if it is then how can this relationship be established? Any insights will be appreciated.
If $ u\in M^{\perp } $ Then we can write $u=u_1+u_2 $ such that $u_1\in M , u_2\in Y$. So $u-u_2=u_1\in M\cap M^{\perp } ={0}$ So $u_1=0$ and $u=u_2\in Y$