For all the problems, let be a positive definite kernel and be the non-empty input space:

- Prove that if
for all , the kernel is identically zero, i.e.
.
- Give an example of a kernel which is positive definite, but not positive in the sense that
for all
. Give an example where the contrary is the case.
- Prove that the inhomogeneous polynomial
with
is a positive definite kernel.
- Give an example of a kernel with two valid feature maps
, mapping into spaces of different dimensions.
- Show that a reproducing kernel (i.e.
for all , where is a Hilbert space of functions
), is symmetric.
- Given a kernel , construct a corresponding normalized kernel by normalizing the feature map
such that for all
. Use this results to show that
is a positive definite kernel in a dot product space .