# How to prove injectivity for kernels?

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.

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.