**Definition:**
Suppose $E$ is a subspace of normed space $X$. Then $E$ is approximately complemented in $X$ if for any compact subset $K$ of $E$ and any $\epsilon>0$ there is a continuous linear operator $P\colon X\to E$ such that $\|x-P(x)\|<\epsilon$ for all $x\in K$.


**Question**: Is there any subspace of a Hilbert space which is not approximately complemented?



Approximately complemented subspaces
The answer is no. That is, any subspace $E$ of a Hilbert space $X$ is approximately complemented. Indeed, take any compact subset $K$ of $E$ and any real $\epsilon>0$.  

Let points $x\_1,\dots,x\_n$ in $K$ form an $\epsilon/2$-net of $K$, and then let $P$ be the orthogonal projector from $X$ onto the linear span of $x\_1,\dots,x\_n$, so that $P$ is a continuous linear operator of norm $\le1$, which may be considered as a map from the Hilbert space $X$ to $E$. Take now any $x\in K$. Then $\|x-x\_i\|<\epsilon/2$ for some $i=1,\dots,n$. Hence,
$$\|x-Px\|\le\|x-x\_i\|+\|x\_i-Px\_i\|+\|P(x\_i-x)\|<
\epsilon/2+0+\epsilon/2=\epsilon. $$ 


