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I am having trouble with exercise 26 in chapter 2 of Peter Petersen's text "Riemannian Geometry." The exercise is stated:

"Using Polarization show that the norm of the curvature operator on $\Lambda^2 T_pM$ is bounded by $|\mathcal{R}|_p \leq c(n)|\text{sec}|_p$ for some constant $c(n)$ depending on dimension and where $|\text{sec}|_p$ denotes the largest absolute value for any sectional curvature of a plane in $T_pM$."

I understand how to write the norm of the curvature tensor as $|R|^2 = R_{ijk}^l R^{ijk}_l$. And I see that \begin{eqnarray*} R_{ljj}^l &=& g(R(\frac{\partial}{\partial x^l},\frac{\partial}{\partial x^j})\frac{\partial}{\partial x^j},\frac{\partial}{\partial x^l})\\ &=& \sum^{n}_1 c \sec(\frac{\partial}{\partial x^j},\frac{\partial}{\partial x^l})\\ &\leq& c(n)|\text{sec}|_p, \end{eqnarray*} where $c$ is some constant, but I'm not sure how to proceed or in what way the author intends us to use polarization...

EDIT (Attempt at a Solution): Since $\mathcal{R}$ is self adjoint, there exists an orthonormal basis for $\Lambda^2 T_pM$ consisting of eigenvectors of $\mathcal{R}$. Let $v_1,...,v_n$ be such a basis, i.e. $\mathcal{R}v_i = \lambda_i v_i, \; g(v_i,v_j) = 0 \; \text{and} \; ||v_i|| = 1$. We prove this using the operator norm defined in the text $|\mathcal{R}|_p = \max\{|\lambda_j|\}$ where $\lambda_j$ is an eigenvalue of $\mathcal{R}.$ Let $|\lambda_k| = \max\{|\lambda_j|\}$ \begin{eqnarray*} |\mathcal{R}|^2_p &=& (\max\{|\lambda_j|\})^2\\ &=& |\lambda_{k}|^2\\ &=& |g(\mathcal{R}(v_k),\mathcal{R}(v_k)|\\ &=& |g(\mathcal{R}(v_k),\lambda_k v_k|\\ &=& |\lambda_{k}||g(\mathcal{R}(v_k), v_k)|\\ &=& |\lambda_{k}| |\sec{(v_k)}|\\ &\leq& |\lambda_{k}| |\sec|_p.\\ \end{eqnarray*} Thus, $|\mathcal{R}|_p = |\lambda_{k}| \leq |\sec|_p$. Then, by equivalence of norms we obtain $|\mathcal{R}|_p| \leq c(n)|\sec|_p$ for the Euclidean norm, where $c(n)$ is some constant $c(n)$ depending on dimension.

The problem with the above is that the $v_i$ might not be simple ($v_i$ might not equal $x_1 \wedge x_2$ where $x_1,x_2 \in T_PM$) so that $g(\mathcal{R}(v_k), v_k)$ isn't a sectional curvature....

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By "polarization" the author understands here the process of computing the sectional curvature $K(X \wedge Y)$ on the plane $ \tfrac{1}{2}(e_i + e_k)\wedge(e_j + e_l) $ that gives an expression of the Riemannian $R_{i j k l}$ in terms of $K(X \wedge Y)$ only! (As it is well known the sectional curvature completely determines the Riemannian curvature)

The full answer can be found, for instance, on p. 23 of the beautiful book of C.Hopper and B.Andrews "The Ricci flow in Riemannian geometry" (can be downloaded from B.Andrews site)

Another reference that may be equally useful (with more information provided) is in Proposition 13.4 on p.404 and subsequent remarks here.

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    I gave it another shot and edited my attempt into the original question.2012-10-22