As an example of an inner product space that is not a Hilbert space, consider the inear space C[a, b] (Example 1.2) with inner product of f, g € C[a, b] defined to be (f. g) = f*f(x)g(x) dx. f a sequence of functions from C[a, b], {f} converges with respect to the norm lefined by this inner product, they will only converge “in the mean!" Since mean convergence does not necessarily imply uniform convergence it is seen that this paces does not have to be complete and the counterexample Example 4.4) shows hat this is actually so. Request explain which is this norm?

Q: 1. Let V be the set of all functions that are continuous on [-1.1]. Let W be the set of all…

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Q: Explain why the Lp[0, 1] spaces are complete normed linear spaces. What is the norn of function f(x)…

A: This is a problem of functional analysis.

Q: . Exercise 1.2.10 Let V₁ and V₂ be normed spaces with norms || ||1 and ||||2. Recall that the…

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Q: (b) Let f be a bounded linear functional on a subspace M of a normed space X and let X, E X, x, 4 M.…

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Q: If X/M ={ [x], xe Xx} show that X/M is a linear space, Where [x] is equivalence class

A: To show that X/M is a linear space, we need to verify that it satisfies the following axioms:…

Q: Define a function f : C -> C by f(x+iy) = (x+2y) + i(3x+4y) for x,y in R. Show that f is additive…

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Q: Show that Lˆ is a linear operator, i.e., that is satisfies the necessary requirement for linear…

A: As per the question we are given a operator L̂ on the Hilbert space over ℂ [a, b] and we have to :…

Q: Let X & Y be normed spaces & x # 204. Then BL (X₂Y) is a Banach space in the operator norm if and…

A: Here BL(X,Y) (or, B(X,Y)), is the space of bounded linear operators from X to Y. First ,we show that…

Q: Q7) (a) Let X be a Banach space and R the usual normed space. Show that ||(z, a)|| = ||||+|al is a…

A: (a) Given a Banach space  and  the usual normed space, we want to show that the norm  is complete on…

Q: 6. Show that (an)neN C R is bounded if and only if lim supn→∞ an < +∞o.

A: To Prove:  ann is bounded sequence if and only if limsupan&lt;+∞.

Q: Consider the Hilbert space H = Let (X, , μ) be a o-finite measure space. L²(X, N, μ) =: L² (µ). If ¢…

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Q: Let T be a linear operator on a finite-dimenstona! inner product space V. Let E1,..., Ex be linear…

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Q: If A is a bounded linear operator on a H, if Hilbert space H, prove that R(A*) = and only if A is…

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Q: (a) Let F be a finite dimensional subspace of an inner product space X. Then show that X=F+ F-and F…

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Q: Let A CR be the set of integers, viewed as a subspace of the reals, and let X be the quotient space…

A: please find the attached image

Q: (a) Formulate and prove the theorem on boundedness and continuity of linear operators on normed…

A: (a) Theorem on Boundedness and Continuity of Linear Operators on Normed Spaces: Let X and Y be…

Q: 2.57 † Let S = functions on [a,b]. Show that if n e N and if ao, a1,..., an are not all 0, then it…

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Q: Show that a bounded linear operator A on complex Hilbert space H is self adjoint if and only if is…

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Q: Consider the operator Î=k, where k is a scalar, operating in a function space on the interval [a, b]…

A: a) To show that L^ is a linear operator, we need to show that it satisfies the following properties:…

Q: Show that Z[V-2] is a Euclidean Domain with respect to the norm N defined by N(a + by-2) = a² + 2b².…

A: This is a problem of abstract algebra.

Q: 3. Consider the set e = {x = (r1, x2, x3, ...) E C* : i=1 consisting of all absolutely summable…

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Q: Show that if a non-zero element in a normed space X, then there is a bounded linear functional f on…

A: We need to show that if a non-zero element in a normed space X, then there is a bounded linear…

Q: is an Let V(F) be an inner product space. If 0 + o e V then prove that orthonormal set.

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Q: 1.Solve the following polynomial equations.  a)  2X3+1=x2+2X

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Q: Let V be a finite-dimensional inner product space over F. (a) Prove that the trace of every positive…

A: Below given clear explanation

Q: .: Remind that the set F of all functions from R to R is a vector space. Prove that the set E of…

A: We know that If W is a subset of a vector space V , then W is a subspace of V if and only if the…

Q: Let V = R[x] = {p(x): grad (p(x) ≤ n} the vector space of the set of polynomials of degree less than…

A: The mapping f:U→V is said to be homomorphism from vector space U to vector space V over the field F…

Q: Verify that the set C[a, b] of all continuous real-valued functions defined on the interval a ≤ x ≤…

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Q: t & · &1 and & · &2 be two equivalent norms on a vector space X. Prove that a sequence % xn & in X…

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Q: 12. Suppose that a, b E R and a < b. Show that the vector space C[a, b] of all continuous complex…

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Q: Let A: X₁ ≤X → Y be a closed operator where X and Y are Banach spaces. Define ||x|| = ||x||+||Ax||,…

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Q: Let n € N and let V be the set of polynomials in the indeterminate x with coefficients in R of…

A: We will be solving part (b), (c), and (d) as mentioned. Given n∈ℕ. Now define:…

Q: Prove that Co is a Banach space by showing that it is a closed subspace of l

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Q: Let X be the vector space of all ordered pa y = (n1, N2), · of real numbers. Show that norms on %3D…

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Q: Let T be a linear operator on a finite-dimenstona! inner product space V. Let E1,..., Ex be linear…

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Q: The vector space C[a, b] of all continuous real-valued functions on [a, b] is not the span of any…

A: A vector space is a set whose elements, often called vectors, may be added together and multiplied…

Q: Let V be an inner product space and T be a linear operator on V with || Tv||= ||v||  ∀ v ∈ V. Show…

A: Let V be an inner product space and T be a linear operator on V with Tv=v ∀v∈V.

Q: Determine whether the given set S is a subspace of the vector space V. A. V is the vector space of…

A: We have to determine whether the given set S is a subspace of the vector space V.(A) V is the vector…

Q: 4. Let W be a subspace of a vector space V. We define a relation: V₁ V₂ if V₁ V₂ € W. ~ 4a. Show…

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Q: Q1. Let V be a vector space over a field (F, +, .), let S={ vi, v2, V3 } be a linearly independent…

A: Let V be vector space over a field (F, +, .).