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We present and discuss the X-ray luminosity function (XLF) of the ROSATBrightest Cluster sample (BCS), an X-ray flux limited sample of clusters ofgalaxies in the northern hemisphere compiled from ROSAT All-Sky Survey data.The BCS allows the local cluster XLF to be determined with unprecedentedaccuracy over almost three decades in X-ray luminosity and provides animportant reference for searches for cluster evolution at higher redshifts. Wefind the significance of evolution in both the XLF amplitude and in thecharacteristic cluster luminosity L_X^star to be less than 1.8 sigma within theredshift range covered by our sample thereby disproving previous claims ofstrong evolution within z < 0.2.Comment: LaTeX using AAS style file, 1 table, 5 PostScript figures. To appear  in the April 20 issue of ApJL (in press). Changes: 1. The code used to fit a  Schechter function to our data did not take the 10% incompleteness of our  sample properly into account. The best fit parameters obtained with the  corrected version differ slightly from the earlier ones. None of our  conclusions are affected. 2. We include a comparison with the XLF obtained by  De Grandi for the BSGC-KP cluster sample

The [ITAL]ROSAT[/ITAL] Brightest Cluster Sample (BCS): The Cluster X-Ray Luminosity Function within [CLC][ITAL]z[/ITAL][/CLC] = 0.3