  We describe recent spectroscopic observations of red giant stars made by the
Space Telescope Imaging Spectrograph (STIS) instrument on board the Hubble
Space Telescope, which have provided spatially resolved observations of the
warm chromospheric winds that predominate for early K to mid-M giants. The H I
Lyman-alpha lines of a set of 11 red giants observed with the STIS/E140M
echelle grating are first analyzed to ascertain wind H I column densities and
total wind mass-loss rates. The M giants have estimated mass-loss rates of
Mdot=(14-86)e-11 Msun/yr, while the K giants with detected wind absorption have
weaker winds with Mdot=(1.5-2.8)e-11 Msun/yr. We use long-slit spectra of H I
Lyman-alpha for two particular red giants, Alpha Tau (K5 III) and Gamma Cru
(M3.5 III), to study the spatial extent of the Lyman-alpha emission. From these
data we estimate limits for the extent of detectable emission, which are r=193
Rstar for Gamma Cru and r=44 Rstar for Alpha Tau. Cross-dispersion emission
profiles in the STIS echelle spectra of the larger sample of red giants also
show evidence for spatial resolution, not only for H I Lyman-alpha but for
other lines with visible wind absorption, such as Fe II, Mg II, Mg I, O I, and
C II. We characterize the nature of these spatial signatures. The spatial
extent is far more apparent for the M giants than for the K giants, consistent
with the stronger winds found for the M giants from the Lyman-alpha analysis.
