In this paper, we mainly study the function spaces related to H-sober spaces. For an irreducible subset system H and T_0 spaces X and Y, it is proved that Y is H-sober iff the function space ℂ(X, Y) of all continuous functions f : X⟶ Y equipped with the topology of pointwise convergence is H-sober iff the function space ℂ(X, Y) equipped with the Isbell topology is H-sober. One immediate corollary is that for a T_0 space X, Y is a sober space (resp., d-space, well-filtered space) iff the

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... space ℂ(X, Y) equipped with the topology of pointwise convergence is a sober space (resp., d-space, well-filtered space) iff the function space ℂ(X, Y) equipped with the the Isbell topology is a sober space (resp., d-space, well-filtered space). It is shown that T_0 spaces X and Y, if the function space ℂ(X, Y) equipped with the compact-open topology is H-sober, then Y is H-sober. The function space ℂ(X, Y) equipped with the Scott topology is also discussed.