Connectedness of topological groups and topological vector spaces

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Problem IV.3.4 from Conway J.B. "A Course in Functional Analysis" [Con07]More problems in Section IV.3 "Some Geometric Consequences of the Hahn-Banach Theorem"

(a)(b)

Let $\mathscr {X}$ be a real TVS. Show:

(a)

if $G$ is an open connected subset of $\mathscr {X}$ , then $G$ is arcwise connected;

(b)

if $f : \mathscr {X} \to \mathbb {R}$ is a continuous non-zero linear functional, then $\mathscr {X} \setminus \ker f$ has two components, $\left\{ x : f(x) > 0\right\}$ and $\left\{ x : f(x) < 0\right\}$ .

Problem IV.3.5 from Conway J.B. "A Course in Functional Analysis" [Con07]More problems in Section IV.3 "Some Geometric Consequences of the Hahn-Banach Theorem"

If $\mathscr {X}$ is a complex TVS and $f : \mathscr {X} \to \mathbb {C}$ is a nonzero continuous linear function, show that $\mathscr {X} \setminus \ker f$ is connected.

Problem 3.5.6 from Rudin W. "Functional Analysis" [Rud91]More problems in Chapter 3.5 "Some Applications"

Suppose $G$ is a topological group and $H$ is the largest connected subset of $G$ that contains the identity element $e$ . Prove that $H$ is a normal subgroup of $G$ , that is, a subgroup that satisfies $x^{-1} H x = H$ for every $x \in G$ . Hint: If $A$ and $B$ are connected subsets of $G$ , so are $A B$ and $A^{-1}$ .

Problem 2.94 from Steeb W., Tanski I., Hardy Y. "Problems and Solutions for Groups, Lie Groups, Lie Algebras with Applications" [STH12]More problems in Chapter 2 "Lie Groups"

Show that the Lie group $U(n)$ is connected.

References

[Con07] Conway J.B. A Course in Functional Analysis. 2nd ed., Springer, 2007.

[Rud91] Rudin W. Functional Analysis. 2nd ed., McGraw-Hill, 1991.

[STH12] Steeb W., Tanski I., Hardy Y. Problems and Solutions for Groups, Lie Groups, Lie Algebras with Applications. World Scientific Publishing, 2012.