1
vote
1answer
42 views

About the continuity of a function in the closed graph theorem proof

I'm reading Functional Analysis book of Rudin, and in the proof of the closed graph theorem, there's one point that I don't understand. Can someone please explain it to me? I really appreciate this. ...
4
votes
1answer
32 views

Question about the continuity property of a function in topological vector space

I'm reading Functional Analysis book of Walter Rudin, and there's one point in this book that I don't know why he states that. Here is the statement: $f$ is a linear mapping from F-space $X$ into ...
2
votes
1answer
152 views

Why are the Differential- and multiplication mapping on $C^{\infty}(\Omega)$ continuous?

Let $\Omega\subset\mathbb{R}^n$ be open and $\Omega\neq\varnothing$ and suppose we have the Fréchet topology on $C^{\infty}(\Omega)$ (this can be obtained by the topology construction from out ...
4
votes
2answers
260 views

Continuous linear functionals

Let L be a continuous linear functional on a metric linear space X. Prove: L(S) is a bounded set for any bounded subset S of X. The metric is translation invariant.
1
vote
2answers
176 views

Left topological zero-divisors in Banach algebras.

Let $ A $ be a unital Banach algebra. Define $ \zeta: A \longrightarrow [0,\infty) $ by $$ \forall a \in A: \quad \zeta(a) \stackrel{\text{def}}{=} \inf_{b \in \mathbb{S}(A)} \| ab \|, $$ where $ ...
1
vote
2answers
281 views

Continuity in Frechet spaces

These are undoubtably simple questions, but I have no background in functional analysis and am wondering about them. The first is an exercise from Folland, the second is not, but both are claims I've ...