test-transclusion-a

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This is transclusion a

This is transclusion B

This is transclusion C

This is trans D

The following canonical map is injective: \[ (\ell^\infty(G) \otimes C(X)) \rtimes_{\max} G \to (\ell^\infty(G)\otimes \ell^\infty(Y;X))\rtimes_{\max} G. \]

one two This is transclusion B

This is transclusion C

This is trans D

The following canonical map is injective: \[ (\ell^\infty(G) \otimes C(X)) \rtimes_{\max} G \to (\ell^\infty(G)\otimes \ell^\infty(Y;X))\rtimes_{\max} G. \]

Let \(V\) be an inner product space and let \(W \leq V\) be a subspace. Let \(v \in V\) and define \(\hat{v} \in W\) as in the proof of Proposition 2.2.3. Prove that if \(w \in W\) with \(w\neq \hat{v}\), then \(\left\| v-\hat{v} \right\| < \left\| v-w \right\|\). Deduce that \(\hat{v}\) is independent of the choice of orthonormal basis for \(W\). It is called the orthogonal projection of \(v\) onto \(W\).

Hint: Use the notation of the proof of Proposition 2.2.3. Let \(\{ e_{m+1},\ldots,e_{n} \}\) be an orthonormal basis for \(W^\perp\). Then \(\{ e_{1},\ldots,e_{n} \}\) is an ONB for \(V\). Say \(v = \sum_{i=1}^n a_i e_{i}\) and \(w = \sum_{i=1}^m b_{i}e_{i}\). To compute the norms of \(v - \hat{v}\) and \(v-w\), express both vectors as linear combinations of \(e_{1},\ldots,e_{n}\).

\(A\nothing\)