I'm new to stochastic calculus, so hopefully this isn't too silly of a question.
Setup:
Let $(\Omega,\mathcal{F},\mathcal{F}_t,\mathbb{P})$ be a filtered probability space on which a Brownian motion $W_t$ is defined (and adapted) and fix $T>0$. Let $\mathcal{L}^2(W)$ denote the Hilbert space of all progressively measurable stochastic processes $\phi$ defined on $(\Omega,\mathcal{F},\mathcal{F}_t,\mathbb{P})$ which satisfy $$ \|\phi\|_{\mathcal{L}^2(W)}\triangleq \mathbb{E}\left[ \int_0^T \phi_s^2 d[W]_s \right]<\infty. $$ Let $H$ denote the Hilbert subspace $$ \left\{ X \in L^2_{\mathbb{P}\otimes m}(\mathcal{F}\times \mathcal{B}([0,\infty)):\, (\exists \phi \in \mathcal{L}^2(W) ) \, X_t = \int_0^{T} \phi_s dW_s \right\} , $$ where $m$ is the Lebesgue measure on $[0,\infty)$.
Question:
Does there exist a bi-Lipschitz function (or even better, a linear isometry) between $L^2_{\mu}([0,\infty))$ and either of these two spaces:
- ${\mathcal{L}^2(W)}$?
- $H$?
Where $\mu$ is a Borel measure, defined on the Borel $\sigma$-algebra $\mathcal{B}([0,\infty))$?