  We prove that for a right linear bounded normal operator on a quaternionic
Hilbert space (quaternionic bounded normal operator) the norm and the numerical
radius are equal. As a consequence of this result we give a new proof of the
known fact that a non zero quaternionic compact normal operator has a non zero
right eigenvalue. Using this we give a new proof of the spectral theorem for
quaternionic compact normal operators.. Finally, we show that every
quaternionic compact operator is norm attaining and prove the Lindenstrauss
theorem on norm attaining operators, namely, the set of all norm attaining
quaternionic operators is norm dense in the space of all bounded quaternionic
operators defined between two quaternionic Hilbert spaces.
