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The HST-TESS Exoplanet Initiative (HTEI) was recently recommended by STScI to capitalize on the wealth of nearby sub-Neptunes discovered by the Transiting Exoplanet Survey Satellite (TESS). Even though sub-Neptunes are very common comma there are still major open questions about how they form and evolve. Direct measurements of their atmospheric composition are a powerful tool to distinguish between different formation models. Fortunately comma the abundance of new TESS discoveries provides an opportunity to systematically study sub-Neptune atmospheres for the first time. Here we propose a community-driven HTEI ProgramdoublePoint SPACE (the Sub-neptune Planetary Atmosphere Characterization Experiment). The SPACE Program will combine UV stellar characterization with WFC3 transmission spectroscopy for eight sub-Neptune systems. The targets span a physically-motivated grid across planet radius (2 - 3.5 Re) and temperature (300 - 1400 K) comma specifically designed to reveal how sub-Neptune atmospheres are shaped by UV photochemistry comma equilibrium cloud formation comma atmospheric metal enrichment comma vertical mixing comma and photoevaporation. The UV spectra are urgently needed because they are a crucial input for photochemistry and atmospheric escape models comma and can only be obtained while HST is still operational. With these measurements comma SPACE will extend HST.s exoplanet legacy from hot Jupiters to the much more abundant sub-Neptunes comma to rapidly establish demographic trends and identify cloud-free atmospheres for future follow-up of this exciting population.