To investigate the fundamental principles of H2 formation in a giant molecular cloud (GMC), we derive the HI and H2 surface density (Sigma_HI and Sigma_H2) images of the Perseus molecular cloud on sub-pc scales (~0.4 pc). We use the far-infrared data from the Improved Reprocessing of the IRAS Survey and the V-band extinction image provided by the COMPLETE Survey to estimate the dust column density image of Perseus. In combination with the HI data from the Galactic Arecibo L-band Feed Array HI Survey and an estimate of the local dust-to-gas ratio, we then derive the Sigma_H2 distribution across Perseus. We find a relatively uniform Sigma_HI ~ 6-8 Msun pc^-2 for both dark and star-forming regions, suggesting a minimum HI surface density required to shield H2 against photodissociation. As a result, a remarkably tight and consistent relation is found between Sigma_H2/Sigma_HI and Sigma_HI+Sigma_H2. The transition between the HI- and H2-dominated regions occurs at N(HI)+2N(H2) ~ (8-14) x 10^20 cm^-2. Our findings are consistent with predictions for H2 formation in equilibrium, suggesting that turbulence may not be of primary importance for H2 formation. However, the importance of a warm neutral medium for H2 shielding, an internal radiation field, and the timescale of H2 formation still remain as open questions. We also compare H2 and CO distributions and estimate the fraction of "CO-dark" gas, f_DG ~ 0.3. While significant spatial variations of f_DG are found, we do not find a clear correlation with the mean V-band extinction.

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