Intereting Posts

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Pre Algebra book Recommendation

Here is the proof of the Riesz Lemma.

When the author show that $\| z-y \| \geq \theta$, I don’t understand the following step:

$$\dfrac{1}{\| v-y_0 \|} \| v-y_0 – (\| v-y_0 \|)y \| \geq \dfrac{\alpha}{\| v-y_0 \|}$$

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How to obtain the above step?

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$v – y_0 – (||v-y_0||)y$ is of the form $v – y’$ with $y’ \in Y$ (as $Y$ is a subspace, and $y, y_0 \in Y$).

The norm of it is thus bounded below by $\alpha$ by the definition of $\alpha$ as $\inf_{y \in Y} ||v – y||$.

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