[proofplan] We first prove the identity for bounded simple predictable processes by expanding the stochastic integral as a finite sum of predictable coefficients times Brownian increments. Independence and zero mean make the cross terms vanish, while the diagonal terms contribute the time increments. We then extend from simple predictable processes to all square-integrable predictable processes by density, defining the Itô integral as the $L^2$ limit and preserving the norm identity by continuity. [/proofplan] [step:Prove the isometry for bounded simple predictable processes] Let $\mathcal S_T$ denote the [vector space](/page/Vector%20Space) of bounded simple predictable real-valued processes on $[0,T]$. Let $H\in\mathcal S_T$ have the form \begin{align*} H_t &= \sum_{i=1}^{m} H_i\,\mathbb 1_{(t_{i-1},t_i]}(t), \end{align*} where $0=t_0<t_1<\cdots<t_m=T$ and each $H_i$ is a bounded $\mathcal F_{t_{i-1}}$-measurable real-valued random variable. Define the [linear map](/page/Linear%20Map) $I_T:\mathcal S_T\to L^2(\Omega,\mathcal F,\mathbb P)$ on this process by \begin{align*} I_T(H) &= \sum_{i=1}^{m} H_i\left(W_{t_i}-W_{t_{i-1}}\right). \end{align*} For $i<j$, the product $H_i(W_{t_i}-W_{t_{i-1}})H_j$ is $\mathcal F_{t_{j-1}}$-measurable, while $W_{t_j}-W_{t_{j-1}}$ is independent of $\mathcal F_{t_{j-1}}$ and has mean $0$. Hence \begin{align*} \mathbb E\left[H_iH_j\left(W_{t_i}-W_{t_{i-1}}\right)\left(W_{t_j}-W_{t_{j-1}}\right)\right]&=0. \end{align*} For the diagonal terms, $H_i$ is $\mathcal F_{t_{i-1}}$-measurable and $W_{t_i}-W_{t_{i-1}}$ is independent of $\mathcal F_{t_{i-1}}$ with variance $t_i-t_{i-1}$, so \begin{align*} \mathbb E\left[H_i^2\left(W_{t_i}-W_{t_{i-1}}\right)^2\right] &= \mathbb E[H_i^2]\,(t_i-t_{i-1}). \end{align*} Expanding the square of $I_T(H)$ and using these two computations gives \begin{align*} \mathbb E[I_T(H)^2] &= \sum_{i=1}^{m}\mathbb E[H_i^2]\,(t_i-t_{i-1}) \\ &= \mathbb E\left[\int_0^{\,T} H_t^2\,d\mathcal L^1(t)\right]. \end{align*} [/step] [step:Extend the integral by completion in the square-integrable predictable norm] Let $\mathcal H_T$ denote the [vector space](/page/Vector%20Space) of predictable real-valued processes $K=(K_t)_{0\leq t\leq T}$ with \begin{align*} \|K\|_{\mathcal H_T}^2 &= \mathbb E\left[\int_0^{\,T} K_t^2\,d\mathcal L^1(t)\right]<\infty. \end{align*} The bounded simple predictable processes are dense in $\mathcal H_T$ by the [Density of Simple Processes](/theorems/2091). Choose bounded simple predictable processes $(H_r)_{r\geq 1}$ such that \begin{align*} \|H_r-H\|_{\mathcal H_T}&\to 0. \end{align*} The simple-process isometry from the preceding step gives \begin{align*} \mathbb E\left[\left|I_T(H_r)-I_T(H_q)\right|^2\right] &= \|H_r-H_q\|_{\mathcal H_T}^2. \end{align*} Thus $(I_T(H_r))_{r\geq 1}$ is a [Cauchy sequence](/page/Cauchy%20Sequence) in $L^2(\Omega,\mathcal F,\mathbb P)$. Since $L^2(\Omega,\mathcal F,\mathbb P)$ is complete, it has a limit. Define \begin{align*} \int_0^{\,T} H_t\,dW_t &:= \lim_{r\to\infty} I_T(H_r) \end{align*} in $L^2(\Omega,\mathcal F,\mathbb P)$. [/step] [step:Pass the isometry identity to the limit] The definition is independent of the approximating sequence. Indeed, if $(K_r)_{r\geq 1}$ is another bounded simple predictable sequence with $\|K_r-H\|_{\mathcal H_T}\to 0$, then \begin{align*} \mathbb E\left[\left|I_T(H_r)-I_T(K_r)\right|^2\right] &= \|H_r-K_r\|_{\mathcal H_T}^2 \\ &\leq \left(\|H_r-H\|_{\mathcal H_T}+\|H-K_r\|_{\mathcal H_T}\right)^2\to 0. \end{align*} Taking limits in the simple-process identity gives \begin{align*} \mathbb E\left[\left(\int_0^{\,T} H_t\,dW_t\right)^2\right] &= \lim_{r\to\infty}\mathbb E[I_T(H_r)^2] \\ &= \lim_{r\to\infty}\mathbb E\left[\int_0^{\,T} (H_r(t))^2\,d\mathcal L^1(t)\right] \\ &= \mathbb E\left[\int_0^{\,T} H_t^2\,d\mathcal L^1(t)\right], \end{align*} where the final equality is convergence in the norm $\|\cdot\|_{\mathcal H_T}$. This proves the Itô isometry. [/step]