If Y is a subspace of a vector space X and f is a linear functional on X such that f(Y) is not the whole scalar field of X. Show that f(y)=0 for all y∈Y.

Suppose that a random sample is to be taken from a normal distribution for which the value of the mean θ is unknown and the standard deviation is 2, and the prior distribution of θ is a normal distribution for which the standard deviation is 1. What is the smallest number of observations that must be included in the sample in order to reduce the standard deviation of the posterior distribution of θ to the value 0.1?