How a Linear problem becomes Non linear in the given situation?

Question

Let $X$ and $Y$ be Banach spaces and $T: X \to Y$ is a linear map. Now originally the problem is find best approximate solution of $$Tx = y$$ Now if i am interested in finding the best approximate solution of above problem in a certain closed and convex subset $C$ of $X$. How this problem becomes non linear due to this constraint?

Answer 1

A problem is called linear if the function is linear and the space (where the function is defined, or where the optimisation happens) is linear. A closed and convex subset does not need to be linear.

An example of convex, closed but nonlinear set is the closed unit ball in $\mathbb{R}^2$.

In general, closed half-spaces are both closed and convex sets. They are not linear.

Since arbitrary intersections of closed sets are closed, and of convex sets are convex, it is easy to construct more by taking intersections of closed half-spaces. In fact, in finite-dimensional real space every closed and convex set is an intersection of a possibly infinite number of half-spaces.

See e.g. https://en.wikipedia.org/wiki/Supporting_hyperplane for images.