Exercise 10, § 1

Topological Vector Spaces, V, p. 61

Let $E$ be a real Hilbert space, $B$ a bilinear form on $E \times E$ , $c_1, c_2$ two numbers $> 0$ such that

\begin{align*} |B(x, y)| &\leq c_1 \|x\| \cdot \|y\| \quad \text{for all } x, y \text{ in } E; \\ |B(x, x)| &\geq c_2 \|x\|^2 \quad \text{for all } x \in E. \end{align*}

Prove that for every continuous linear form $f$ on E, there exists a unique element $x_f \in E$ (resp. $y_f \in E$ ) such that $f(x) = B(x_f, x)$ (resp. $f(x) = B(x, y_f)$ ) for all $x \in E$ .

Solution

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