Here is an infinite-dimensional version of Euclidean space: In the space of all infinite sequences, consider the subspace ℓ2 of square-summable sequences [i.e., those sequences (x1, x2, …) for which the infinite series x1^2 .x2^2 ⋯ converges ] . For x⃗ and y⃗ in ℓ2, we define x⃗=√(x1^2 x2^2 ⋯), x⃗ ·y⃗=x1 y1 x2 y2 ⋯(Why does the series x1 y1 x2 y2 ⋯converge?) a. Check that x⃗=(1, (1)/(2), (1)/(4), (1)/(8), (1)/(16), …) is in ℓ2, and find x⃗. Recall the formula for the geometric series: 1 a a^2
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Here is an infinite-dimensional version of Euclidean space: In the space of all infinite sequences, consider the subspace ℓ2 of square-summable sequences [i.e., those sequences (x1,
VIDEO ANSWER: Here is an infinite-dimensional version of Euclidean space: In the space of all infinite sequences, consider the subspace \ell_{2} of square-summ…
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