For the purposes of our discussion, a Space is defined by the image bands. Each band defines one dimension of the space. Thus if we have two spectral bands they define a two dimensional space. This space can be visualized by plotting the two spectral intensities for each pixel in a 2 dimensional x-y plot such as the figure below which is a plot of bands 1 and 2 from the Morro Bay scene.

If we add another band, then we will have a 3 dimensional space. We can still visualize this and even make a 3D model (as shown below with blue data points on the tips of red pins to show location on the Band 1 - Band 2 plane) but when we add the fourth band, we must resort to arguments by analogy to the lower dimensional 2D and 3D spaces.

With the 7 spectral bands available from Thematic Mapper, a 7 dimensional space is defined by the observations.