This might be very stupid question, but I just realized that I haven't spotted a definition for a neighborhood in normed linear space.
Usually in topology one just states that neighborhood is open set of some space such that your point belongs into that open set.
But what if we are looking into normed linear spaces?
What is a neighborhood of a functional?
For example let $ \mathcal{R}$ be a normed linear space and take a point $y$ from it. What would be the neighborhood of this $y$?
An "inner product", , on a vector space automatically defines a "norm" by $||u||= \sqrt{<u,u>}$. An inner product, ||v||, automatically defines a "metric" by d(u,v)= ||u- v||. And a metric then automatically defines a topology by defining open sets in terms of unions and finite intersections of "neighborhoods" $N(p,\rho)= \{q| d(p,q)<\rho\}$.