Continuous function of GL(n,C) I was reading a book on Lie groups where they define a function $f: GL(n,C)->GL(n,C)$, by
$f(B)=B^\dagger B$
where $B \in GL(n,C)$, and $GL(n,C)$ denotes the general linear group over C (complex field). 
They mention something like "it's easy to see $f$ is continuous". It doesn't seem very obvious to me, is there a simple way to see this? I am not very fluent in topology so I apologize if this question is very trivial.
Thanks in advance.
 A: 1 .  The topology on $G = \textrm{GL}_n(\mathbb{C})$  is the subspace topology coming from $\mathbb{C}^{n^2}$, the $n^2$-fold product of copies of $\mathbb{C}$.
2 .  If $Z$ is a subspace of a topological space $Y$, then a function $f: X \rightarrow Z$ is continuous if and only if $f$ is continuous as a map from $X$ to $Y$.  This follows directly from the definition of the subspace topology.  Similarly, if $S$ is a subspace of $X$, and $f: X \rightarrow Y$ is a continuous function, then the restriction of $f$ to $S$ is a continuous function from $S$ to $Y$.
3 .  If $Y$ is a topological space, and $n$ is an integer, let $Y^n$ be the $n$-fold product of copies of $Y$.  Let $\pi_i: Y^n \rightarrow Y$ be the function $(y_1, ... , y_n) \mapsto y_i$.  A function $f: X \rightarrow Y^n$ is continuous if and only if each $\pi_i \circ f: X \rightarrow Y$ is continuous.  This follows directly from the definition of the product topology.
Using these principles, you can reduce the problem to verifying that certain functions $\mathbb{C}^{n^2} \rightarrow \mathbb{C}$ are continuous.
