$ {L}_{1} $ Norm Regularized Least Squares - Strict Convexity Property My objective function that is to be minimized is as follows:
$$\|y-Ax\|_2^2 + \alpha\|Lx\|_1$$
where $L$ is the gradient operator.
Now this problem seems convex because the first term is quadratic and hence convex and the second is a norm so it has to be convex (the triangular inequality), but is it also strictly convex? and why?

I may be wrong but Strictly Convex is also related to the Norm of the Space.  
Assuming Euclidean Space, the Quadratic Function, Given $ A $ is full rank (Its Null Space contains only the trivial $ \boldsymbol{0} $ vector), is Strictly Convex (And Strongly Convex).  
If I remember correctly, a sum of Strictly Convex and Convex Function is a Strictly Convex function.
