How transmission speed and bandwidth are linked?
Actually you also have to consider the noise floor of the communication channel and transmitted signal power to define the limit of data rate through an AWGN channel. The information capacity $C$ (bps) of the channel is given by the Shannon's formula:

$$ C = B \cdot \log_2( 1 + \frac{ \sigma_x^2}{ \sigma_n^2 } ) $$ 

where $B$ is the channel bandwidth in Hz., and $\frac{ \sigma_x^2}{ \sigma_n^2 }$ is the channel SNR (Signal to Noise Power Ratio).

One ideal case happens when there's no noise, then the capacity goes to infinity. This means that on a noiseless (ideal) channel you would transmit all the information (and more) instantly.