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\[
\begin{cases}
rf = \frac{\partial f}{\partial t} + r x \frac{\partial f}{\partial x} + \frac{1}{2} \sigma^2 x^2 \frac{\partial^2 f}{\partial x^2}\\
f(T,x) = (x-K)^{+}
\end{cases}
\]
Suppose \(\varphi\) is a bounded linear functional on a Hilbert space V. Then there exists a unique \(h \in V\) such that $$ \varphi(f) = \langle f, h \rangle$$ for all \(f \in V\). Furthermore, \(\Vert \varphi \Vert = \Vert h \Vert\).