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Let $\mathcal{G}$ be the first Gaussian chaos of a centered Gaussian random variable in a Banach space. By Fernique's theorem, $Sg=\mathbb{E}[Xg]$ is a Bochner integral in $B$. The map $S$ is injective, because $Sg=0$ implies $\mathbb{E}[g\ell (X)]=0$ for every continuous linear functional $\ell$, hence $g=0$. Its range becomes a Hilbert space under $⟨Sg,Sv{⟩}_H=\mathbb{E}[gv]$. The embedding in $B$ is continuous by Cauchy-Schwarz inequality. Its reproducing property is $⟨h,S\ell (X){⟩}_H=\ell (h)$. Thus this is the Reproducing-kernel Hilbert space of the random variable, including degenerate laws. For $h=Sg$, the scalar coordinate $\hat{h}(X)=g$ has variance $\Vert h{\Vert }_H^2$; it is generally a measurable linear coordinate rather than a continuous functional on $B$.

A Gaussian random variable in a Banach space taking values in a separable Banach space satisfies the displayed exponential-integrability bound for some positive $a$. In particular every finite power of its norm has a finite expected value. The value of $a$ depends on the Gaussian measure; the theorem does not assert this for every positive $a$.

For a centered Gaussian random variable in a Banach space, its first Gaussian chaos is the closed linear subspace of L2 space generated by its continuous linear observations. Every element is a centered Gaussian random variable, and every finite collection has a multivariate normal distribution: approximate in L2 space and pass to the characteristic function. The first Gaussian chaos supplies the scalar coordinates of its Cameron-Martin space of a Gaussian random variable in a Banach space.

The Cameron-Martin space of a Gaussian random variable in a Banach space is most conveniently defined using the first Gaussian chaos. Work over the real separable Banach space $B$, with continuous dual space $B^{\ast }$. For a centered Gaussian random variable in a Banach space, defineThe set inside the closure is already a linear subspace. Every $g\in \mathcal{G}$ is a centered Gaussian random variable. The Bochner integral defining $S$ exists: Fernique's theorem states that for some $a>0$, $\mathbb{E}{e}^{a\Vert X{\Vert }_B^2}<\infty$, and then Cauchy-Schwarz inequality gives $\mathbb{E}\Vert Xg{\Vert }_B<\infty$.

The map $S$ is injective. Indeed, $Sg=0$ implies $\mathbb{E}[g\ell (X)]=\ell (Sg)=0$ for all $\ell \in B^{\ast }$. Density of these observations in $\mathcal{G}$ then implies $\mathbb{E}{g}^2=0$. Define the Reproducing-kernel Hilbert space byThe injectivity makes the inner product unambiguous, and the completeness of $\mathcal{G}$ makes $H$ a Hilbert space. The embedding into $B$ is continuous sinceFor $\ell \in B^{\ast }$ put $Q\ell =S(\ell (X))$. The reproducing property isThis defines the Banach-space Gaussian RKHS even for a degenerate Gaussian measure.

For $h=Sg\in H$, write $\hat{h}(X)=g$. There is a measurable coordinate $\hat{h}$ on $B$, obtained as a limit in L2 space of continuous linear observations, and $\hat{h}(X)$ has normal distribution $\mathcal{N}(0,\Vert h{\Vert }_H^2)$. The Cameron-Martin theorem for a Gaussian measure, in its real separable Banach space form, states that if $\mu$ is the probability law of $X$, then the probability law ${\mu }_h$ of $X+h$ is an equivalent probability measure to $\mu$ precisely when $h\in H$. For such a shift,For $h\notin H$ the two laws are mutually singular measures. The coordinate $\hat{h}$ need not be a continuous functional on $B$, and the sample $X$ need not belong to $H$.

Apply this theorem to the shift $-h$. With $a=\Vert h{\Vert }_H^2/2$, it givesChoose the measurable coordinate so that $\hat{h}(-x)=-\hat{h}(x)$ on a symmetric set of full $\mu$-measure. This is possible by taking an almost-surely convergent subsequence of the approximating linear observations and intersecting its convergence set with its negative. Alternatively the joint law of $(X,\hat{h}(X))$ is invariant under simultaneous negation, directly from those observations and their mean-square convergence. Since $C=-C$ and the centered Gaussian measure is symmetric, the two exponential integrals with signs $+$ and $-$ agree. Consequentlybecause $\cosh v\ge 1$. The symmetric Gaussian translation lower bound is thereforeOnly symmetry and Borel measurability of $C$ are required; convexity is not needed.

There is one degenerate flaw in the last printed claim. The separable Banach space $B=\{0\}$ with $X=0$ has $H=\{0\}$ dense in $B$, and all positive-radius balls have probability one. Nevertheless $\Phi (t)=1$ for every $t>0$. Thus the statement needs the additional hypothesis $B\ne \{0\}$. If the course convention excludes the zero space, this hypothesis is already implicit.

Under this necessary hypothesis, density of $H$ implies that $H$ contains a nonzero vector. Given $0<a<b$, scale that vector to obtain $h\in H$ with $\Vert h{\Vert }_B=(a+b)/2$, and choose $0<\delta <(b-a)/2$. The triangle inequality givesThe centered closed ball $C_{\delta }=\{x:\Vert x{\Vert }_B\le \delta \}$ is a symmetric Borel set. Apply the symmetric Gaussian translation lower bound to getHence the Gaussian norm distribution is strictly increasing when $B$ is nonzero. This proves the strict increase of a Gaussian norm distribution without assuming that its distribution has a density or that spheres have zero probability.

For a centered Gaussian random variable in a Banach space, a Borel set $C=-C$, and $h$ in its Cameron-Martin space of a Gaussian random variable in a Banach space, the Cameron-Martin theorem for a Gaussian measure gives the translated density $e^{-\hat{h}(X)-\Vert h{\Vert }_H^2/2}$. The joint law of $(X,\hat{h}(X))$ is invariant under simultaneous negation. Averaging the two density formulas therefore replaces $e^{-\hat{h}(X)}$ by $\cosh \hat{h}(X)\ge 1$, proving the lower bound. No convexity of $C$ is required.