We know that the linear span of subspace $A$ in Hilbert space $X$ is dense if orthogonal complement of $A$, that is $A^{\bot}$ is trivial. I am given a family of functions $A=\{f_n\}$ and I was trying to construct $A^{\bot}$ with just one function, say $g(x)$ so that $g$ is orthogonal to all $f_n$, so that I could conclude that $A$ is not dense in X. But I ended up in constructing a family of functions $B=\{g_n\}$ with $g_n$ and$f_n$ pairwise orthogonal,that is $<f_n,g_n>=0$ for each $n$. My $g_n$'s heavily depend on $f_n's$ for each $n$.
I cannot conclude $A$ is dense, right? From the collection $B$, can I just construct a one function say $g(x)$ which is orthogonal to all of $f_n$?
Based on the data you have given, you cannot conclude that $A$ is dense in $A$. Because if $\{f_n\}$ is an orthogonal family in $X$, then letting $g_n=f_{n+1}$, we would have $\langle f_n,g_n\rangle=0$ for all $n$, but $\operatorname{span}\{g_n\}\subset\operatorname{span}\{f_n\}$.