On the importance of studying Bounded Linear Maps

Question

I have been thinking for some time about the importance of studying boundedness of linear maps on (topological) vector spaces. A few ideas that I have at the moment are as follows:

1. On normed spaces, boundedness and continuity are equivalent. So, if we know that a linear map is bounded, then we can approximate the images of "bad" functions by "good" functions, if we have such density conditions. One example that comes to find is that if we have a bounded linear map $T$ on $L^1(\mathbb{R})$, then for any $\phi \in L^1(\mathbb{R})$, we can approximate $T\phi$ by using images of compactly supported smooth functions. This example comes to me because if $T$ is an integral transform, then compactly supported smooth functions are much easier to work with than general $L^1$-functions, and hence can be termed as "good" functions.

2. Boundedness also tells how much the unit ball is stretched, and consequently any object is stretched. So, it seems that bounded linear maps are important to understand the "scaling-ratio" of these transforms.

Are there any other points of importance while studying bounded maps? Also, from what I have described, the importance seems to be of mathematical interest alone. What applicability does the boundedness of a linear map have?

Answer 1

In general, there are many more theorems and tools that work on continuous functions, so if we are in a normed space, any way we have of dealing with bounded operators will give us a lot more to work with. A big place this shows up is in differential equations: The derivative operator is pretty trivially unbounded, if you look at the space of real valued, differentiable functions on $[0,1]$ under the $\sup$ norm, the family of functions

$$f_n=x^n$$ all have norm 1, whereas their derivative $$nx^{n-1}$$ has norm $n$, thus this is unbounded. However, the integral operator is bounded, since differentiable functions are continuous and continuous functions on a compact set take on extreme values, we have $$\|\int_0^1f(x)dx\|\leq \|f\|$$

Many differential equations can be written in an equivalent integral form, and differential equations show up all over the sciences. So by being able to work with the bounded/continuous version of the equation, we get all of our nice extra tools to work with.