The given function $f(x,y)=x+y$
I have to show that this function maps open set to open set
Ssolution i tried– Here we have to show that $f$ maps open set to open sets which implies that $f$ is an open map.I am trying to prove the answer from backward direction .According to open mapping theorem in functional analysis states that
every surjective continuous linear operator between Banach spaces is an open map
here given given $\mathbb{R^2}$ and $\mathbb{R}$ are both Banach spaces,thus i have to prove that $f$ is linear operator and surjective
$$f(a(x_1,y_1)+b(x_2,y_2))=f(ax_1+bx_2,ay_1+by_2)$$
$$\Rightarrow f(ax_1+bx_2,ay_1+by_2)=ax_1+bx_2+ay_1+by_2 $$
$$\Rightarrow f(ax_1+bx_2,ay_1+by_2)=a(x_1,y_1)+b(x_2,y_2) $$
$$\Rightarrow f(ax_1+bx_2,ay_1+by_2)=af(x_1,y_1)+bf(x_2,y_2)$$
Thus from this we can say that $f$ is a linear functional ,now i have to prove that $f$ is surjective and this is the point where i stuck please give me hint how can i prove that $f$ is surjective.
Thankyou
Best Answer
You are using a very powerful result to prove something trivial. Just prove that the image is open with the definition. For every point in the image, find a nbhd still in the image. Same with density, show that given any point in $\mathbb{R}$ you can get as close as you want with images of elements of a dense set in the plane. By the way, surjectivity is trivial by taking $f(x,0)=x$
P.S. I mean, you’re not wrong, I just like to prove things directly when I can, instead of using powerful abstract tools from functional analysis