Question

Let V1,,Vk\mathbf{V}_{1}, \cdots, \mathbf{V}_{k} be mutually orthogonal vectors in RnR^n. Show that, for any XX in RnR^n, j=1k(XVj)2X2\sum_{j=1}^{k}\left(\mathbf{X} \cdot \mathbf{V}_{j}\right)^{2} \leq\|\mathbf{X}\|^{2}. This is known as Bessel’s inequality for vectors. A version for Fourier series and eigenfunction expansions will be seen in Chapter Fifteen. Hint Let Y=Xj=1k(XVj)VjY = \mathbf{X}-\sum_{j=1}^{k}\left(\mathbf{X} \cdot \mathbf{V}_{j}\right) \mathbf{V}_{j} and compute Y2||Y||^2.

Solution

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REMARK:{\bf REMARK:} In order to obtain the Bessel inequality in given form, we must take normalized vectors!!!

Let us start with a vector Y=Xj=1k(XVj)Vj\boldsymbol{Y}=\boldsymbol{X}-\sum_{j=1}^{k}(\boldsymbol{X}\cdot \boldsymbol{V_j})\boldsymbol{V_j}. Its norm is:

Y2=YY=(Xi=1k(XVi)Vi)(Xj=1k(XVj)Vj)==XXi=1k(XVi)(ViX)j=1k(XVj)(XVj)++i,j=1k(XVi)(XVj)(ViVj)==X2j=1k(XVj)2.\begin{align*} ||\boldsymbol{Y}||^2&=\boldsymbol{Y}\cdot\boldsymbol{Y}=\left(\boldsymbol{X}-\sum_{i=1}^{k}(\boldsymbol{X}\cdot \boldsymbol{V_i})\boldsymbol{V_i}\right)\cdot \left(\boldsymbol{X}-\sum_{j=1}^{k}(\boldsymbol{X}\cdot \boldsymbol{V_j})\boldsymbol{V_j}\right)=\\ &=\boldsymbol{X}\cdot\boldsymbol{X}-\sum_{i=1}^{k}(\boldsymbol{X}\cdot \boldsymbol{V_i})(\boldsymbol{V_i}\cdot \boldsymbol{X})-\sum_{j=1}^{k}(\boldsymbol{X}\cdot \boldsymbol{V_j})(\boldsymbol{X}\cdot \boldsymbol{V_j})+\\ &+\sum_{i,j=1}^k (\boldsymbol{X}\cdot \boldsymbol{V_i}) (\boldsymbol{X}\cdot \boldsymbol{V_j})(\boldsymbol{\boldsymbol{V_i}}\cdot \boldsymbol{V_j})=\\ &=||\boldsymbol{X}||^2-\sum_{j=1}^{k}(\boldsymbol{X}\cdot \boldsymbol{V_j})^2. \end{align*}

The norm is nonnegative, hence

X2j=1k(XVj)20,||\boldsymbol{X}||^2-\sum_{j=1}^{k}(\boldsymbol{X}\cdot \boldsymbol{V_j})^2\geq 0,

i.e.

j=1k(XVj)2X2.\sum_{j=1}^{k}(\boldsymbol{X}\cdot \boldsymbol{V_j})^2\leq ||\boldsymbol{X}||^2.

Remark that

i,j=1k(XVi)(XVj)(ViVj)\sum_{i,j=1}^k (\boldsymbol{X}\cdot \boldsymbol{V_i}) (\boldsymbol{X}\cdot \boldsymbol{V_j})(\boldsymbol{\boldsymbol{V_i}}\cdot \boldsymbol{V_j})

denotes the sum over all ii and jj from 11 to kk; because of the orthogonality of Vi\boldsymbol{V_i} and Vj\boldsymbol{V_j} for iji\neq j this sum over two indices reduces to ordinary sum over one index.

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