  This paper reports the first study of the O VI resonance line emission (1032,
1038 Angstroms) originating in the Local Bubble (or Local Hot Bubble)
surrounding the solar neighborhood. In spite of the fact that O VI absorption
within the Local Bubble has been observed, no resonance line emission was
detected during our 230 ksec Far Ultraviolet Spectroscopic Explorer observation
toward a ``shadowing'' filament in the southern Galactic hemisphere. As a
result, tight 2 sigma upper limits are set on the intensities in the 1032 and
1038 Angstrom emission lines: 500 and 530 photons cm^{-2} s^{-1} sr^{-1},
respectively. These values place strict constraints on models and simulations.
They suggest that the O VI-bearing plasma and the X-ray emissive plasma reside
in distinct regions of the Local Bubble and are not mixed in a single plasma,
whether in equilibrium with T ~ 10^6 K or highly overionized with T ~ 4 to 6 x
10^4 K. If the line of sight intersects multiple cool clouds within the Local
Bubble, then the results also suggest that hot/cool transition zones differ
from those in current simulations. With these intensity upper limits, we
establish limits on the electron density, thermal pressure, pathlength, and
cooling timescale of the O VI-bearing plasma in the Local Bubble. Furthermore,
the intensity of O VI resonance line doublet photons originating in the
Galactic thick disk and halo is determined (3500 to 4300 photons cm^{-2} s^{-1}
sr^{-1}), and the electron density, thermal pressure, pathlength, and cooling
timescale of its O VI-bearing plasma are calculated. The pressure in the
Galactic halo's O VI-bearing plasma (3100 to 3800 K cm^{-3}) agrees with model
predictions for the total pressure in the thick disk/lower halo. We also report
the results of searches for other emission lines.
