Let $X$ be the vector space of all finite linear combinations of functions of the form $e^{i\lambda t}$ ($-\infty<t<+\infty$), where the parameter $\lambda$ is real. An inner product in $X$ is defined by $$(f,g)=\lim_{T\rightarrow \infty}\frac{1}{2T}\int_{-T}^T f(t) \bar g(t) dt.$$ When $X$ is closed by means of the metric generated by this inner product, we obtain a Hilbert space $B^2$. Show that the continuum of elements $e^{i\lambda t}$ forms a complete orthonormal subset of $B^2$.
This is an exercise by
Young, Robert M. "An Introduction to Non-Harmonic Fourier Series", Revised Edition, 93. Academic Press, 2001.
I do not know how to solve this exercise. Any suggestions please?