(i) Let v, v' E V. Suppose that (vlr) = (vr) for every x V. Prove that v = v'. (ii) Suppose that S: V→W and T: W→ V are mappings that satisfies (Sv|w) = (v|Tw) (v EV, WEW) Note that the leff-hand side is the inner product in W (both Su and w are vectors in W), while the right-hand side is the inner product in V (both v and Tw are vectors in V). Prove that S is linear. 1

Q: 2) Let V₁, V2, W be vector spaces over F. Show that the set Bil(V₁ × V₂, W) of bilinear maps is a…

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Q: Prove directly that the dual space of a normed space is complete.

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Q: 3. Prove that, for a given n, all n-dimensional vector spaces with real scalars are isomorphic.

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Q: Which of the listed mappings is an isomorphism of vector spaces?

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Q: Every T3 space is regular space. True False

A: Since a regular T1 space is called a T3 space.

Q: Show that the spaces (R,U) and (R,S) are Hausdorff.

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Q: Suppose X and Y are normed spaces and X + 0. Prove that BL(X, Y) is a Banach space with the operator…

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Q: Let V be an inner product space. Show that ||cu|| |c| ||u|| for any vector u and any scalar c.

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Q: s an inner product space, where (a, 5) = 4 – 5i.

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Q: If S be a n-dimen Sional normed Space, then prove that its dual space is al so n- dimen Sional .

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Q: Let U = span((1, 1, −3, −3),(2, −2, −2, 2)) and W = span((1, 0, 2),(3, 1, 1),(9, 4, −2),(−7, −3,…

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Q: prove that any finite dimensional normed space is reflexive.

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Q: 2) Let V₁, V2, W be vector spaces over F. Show that the set Bil(V₁ x V₂, W) of bilinear maps is a…

A: Please check step 2 and 3 for solution.! We will verify axioms of vector space.

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Q: be a n-dimensional normed space, then prove that its dual space is also n-dimensional.

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Q: Question: For an inner product space V, show that two vectors V₁, V₂ € V are equal if and only if…

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