Assume that $V$ is a complex Banach space, $\mathcal{L}(V)$ is the Banach algebra composed of bounded linear operators with the norm $\lVert A\rVert=\sup\{\lVert Av\rVert:\lVert v\rVert\leq 1 \}$.
$$ \mathcal{L}(V)^* :=\{A\in \mathcal{L}(V):\text{$A$ is invertible in $\mathcal{L}(V)$} \}.$$
Here, I wonder if $\mathcal{L}(V)^*$ is path-connected?
In fact, when $V=\mathbb{C}^n$, $\mathcal{L}(V)^*=GL_n(\mathbb{C})$ is path conneced. So how about a more general situation?
The question has been edited and this answer is no longer valid.
For any normed linear space of dimension a least $2$, $X$ $\{x\in X: x \neq 0\}$ is path connected. In fact, given $x,y \neq 0$ we can find a path within the the two dimensional space spanned by $x$ and $y$ which connects $x$ to $y$.