Let E be a Banach space and \(A\) a bounded linear operator on \(X\). Then for \(x\in A\) the local spectral radius \(r(A,x)\), of A with respect to \(x\) is defined by \[ r(A,x):=\overline{\lim}_{n\to \infty}\| A^ nx\|^{1/n}. \] Clearly, \(r(A,x)\leq r(A)\), the spectral radius of \(A\). The author proves that \(r(A,x)=r(A)\) on a set of second category in \(X\). The convergence of the sequence \((\| A^ nx\|^{1/n})_{n\in {\mathbb{N}}}\) is also discussed.