The task of an extremum seeking controller is to drive or to command a system in order to extremize the value of a performance function that only depends on the present output of the system. Here, an approach for designing such a controller for cases in which the performance function is fully determined by the value of a multi-dimensional parameter is presented. The designed controller will be indeed adaptive; it will estimate the (unknown) value of the parameter, determining the performance function, and accordingly it will issue a command to the system in order to drive it to an output value that extremize the performance function. If a dither function can be found to satisfy some persistent exitation condition, then the output of the system will converge to a neighborhood of the extremizing output value.