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I'm trying to understand the proof of Theorem 4.21 in Support Vector Machines by Ingo Steinwart and Andreas Christmann.

The theorem states that for each kernel $k$ (defined as a function $k:X\times X \rightarrow H_0$ such that $k(x_1,x_2)=\langle\Phi(x_1),\Phi(x_2)\rangle_{H_0}$ for a Hilbert space $H_0$ and map $\Phi:X\rightarrow H_0$) there is a unique reproducing kernel Hilbert space $H$ whose reproducing kernel is $k$.

According to the theorem $$H = \{f : X \rightarrow {\rm I\!R}\mid (\exists w \in H_0)(\forall x \in X)f(x)=\langle w, \Phi(x)\rangle_{H_0}\}$$ and its norm is $$||f||_H=\inf_{w\in H_0}\{||w|| : f = \langle w, \Phi(\cdot) \rangle\}$$

If we define $V:H_0 \rightarrow H$ as follows: $$Vw = \langle w, \Phi(\cdot) \rangle_{H_0} $$ we can conclude that the null space of $V$, $\mathcal{N}(V)$ is closed, hence $H_0$ can be decomposed as $$H_0 = \mathcal{N}(V) \oplus \hat{H}$$ where $\hat{H} = \mathcal{N}(V)^T$. Now, the proof continues by asserting that the restriction $\hat{V}$ of $V$ to $\hat{H}$ is injective by construction, but it's not that obvious to me. How can it be proven?

I see that for any $f_1=V \hat{w}_1,f_2 = V \hat{w}_2 \in H$ ($\hat{w}_1,\hat{w}_2 \in \hat{H})$, if $f_1 = f_2$ then $$V \hat{w}_1 = V \hat{w}_2 \implies V(\hat{w}_1-\hat{w}_2)=0_H \implies \langle \hat{w}_1-\hat{w}_2, \Phi(\cdot)\rangle = 0$$ so if $\Phi$ is surjective, then $\hat{w}_1-\hat{w}_2=0_{H_0} \implies \hat{w}_1=\hat{w}_2$, but the surjectivity of $\Phi$ is not an assumption of the theorem.

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Since $\hat{H}$ is the orthogonal complement of $\mathcal{N}(V)$, it holds that $\hat{V}\hat{w} \neq 0_{H}$ for all $\hat{w}\in\hat{H}$ different from $0_{H_0}$. Therefore, $\mathcal{N}\left(\hat{V}\right)=\{0_{H_0}\}$, so $\hat{V}$ is injective.

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