The Ultracool SpeXtroscopic Survey. I. Volume-Limited Spectroscopic Sample and Luminosity Function of M7$-$L5 Ultracool Dwarfs

We present a volume-limited, spectroscopically-verified sample of M7$-$L5 ultracool dwarfs within 25\,pc. The sample contains 410 sources, of which $93\%$ have trigonometric distance measurements ($80\%$ from \textit{Gaia} DR2), and $81\%$ have low-resolution ($R\sim120$), near-infrared (NIR) spectroscopy. We also present an additional list of 60 sources which may be M7$-$L5 dwarfs within 25\,pc when distance or spectral type uncertainties are taken into account. The spectra provide NIR spectral and gravity classifications, and we use these to identify young sources, red and blue $J-K_S$ color outliers, and spectral binaries. We measure very low gravity and intermediate gravity fractions of $2.1^{+0.9}_{-0.8}\%$ and $7.8^{+1.7}_{-1.5}\%$, respectively; fractions of red and blue color outliers of $1.4^{+0.6}_{-0.5}$\% and $3.6^{+1.0}_{-0.9}$\%, respectively; and a spectral binary fraction of $1.6^{+0.5}_{-0.5}\%$. We present an updated luminosity function for M7$-$L5 dwarfs continuous across the hydrogen burning limit that agrees with previous studies. We estimate our completeness to range between $69-80\%$ when compared to an isotropic model. However, we find that the literature late-M sample is severely incomplete compared to L dwarfs, with completeness of $62^{+8}_{-7}\%$ and $83^{+10}_{-9}\%$, respectively. This incompleteness can be addressed with astrometric-based searches of ultracool dwarfs with \textit{Gaia} to identify objects previously missed by color- and magnitude-limited surveys.