The authors deal with $2(n+m)$ dimensional Hamilton space of the first order and consider linear connections. The general linear connection has $4^{3} = 64$ types of connection coefficients. Different special kinds of linear connection, as almost $d$-connections, $d$-connection, strongly distinguished and almost strongly distinguished connections are defined. The transformation law of connection coefficients are determined. Different covariant derivatives which transform as tensors are obtained. For mentioned different kind of covariant derivatives the torsion and curvature tensors are calculated.