I'm very slowly trying to work my way through these notes on obstruction bundle gluing, section 2.1.
One exercise I want to do is show that $L^2(\mathbb{R}, u^{*}TX)$ is a Banach space and $L^2_1(\mathbb{R}, u^{*}TX)$ is a Banach manifold.
To give more context, let $\pi: TX\to X$ be a tangent bundle and $u: \mathbb{R}\to X$ is a smooth curve. Then $u^{*}TX$ is the pullback of the tangent bundle. $L^2(\mathbb{R}, u^{*}TX)$ is the completion of the space of sections $f: \mathbb{R}\to u^{*}TX$.
I know I need a norm for any element $f\in L^2(\mathbb{R}, u^{*}TX)$, we need $\mid\mid f\mid\mid_{L^2}$ to make sense. But since the codomain is a Banach space (I think?), I'm not sure how to make $||f(s)||_{L^2}^2= \int |f(s)|^2ds$ make sense.
