# QPT

 Apr 17 accepted Relation between range and kernel of a linear operator Apr 17 asked Relation between range and kernel of a linear operator Apr 17 accepted Common eigenvector of linear operator Apr 16 asked Common eigenvector of linear operator Apr 15 awarded Scholar Apr 15 accepted Limit of Inner Products in Hilbert Space Apr 15 comment Limit of Inner Products in Hilbert SpaceFix an $\epsilon$. Finite sums of basis vectors can approximate $g$ to any precision, that is, there exists $\{g_{n}\}_{n = 1}^{\infty}$ and an $N$ such that $\|g_{N} - g\| < \epsilon$. Then $\lim_{k \rightarrow \infty}(f_{k}, g_{N}) = (f, g_{N})$. We also have have that $|(f_{k}, g_{N} - g)| \leq \|f_{k}\|\|g_{N} - g\| < B\epsilon$. Therefore $\lim_{k \rightarrow \infty}(f_{k}, g_{N}) = \lim_{k \rightarrow \infty}(f_{k}, g)$. Then $$(f, g_{N}) = \lim_{k \rightarrow \infty}(f_{k}, g_{N}) = \lim_{k \rightarrow \infty}(f_{k}, g).$$ Is this the right track? Apr 14 awarded Student Apr 14 asked Limit of Inner Products in Hilbert Space