Consider the space , define
. Show that f is a linear functional which is not continuous w.r.t the norm
We will go through the proof step by step. ### Step 1: Understanding \( c_{00} \) The space \( c_{00} \) consists of all sequences \( x = (x_1, x_2, \dots) \) that have only finitely many nonzero terms. That is, there exists some \( N \) such that \( x_n = 0 \) for all \( n > N \). ### Step 2: Checking Linearity of \( f \) The functional \( f: c_{00} \to \mathbb{R} \) is defined as: ___ ______ ________ ____ _________ _____ _________ ______ _________ _________.
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Let X be an inner product space and x, y ∈ X. Prove that x | y if and only if
Define the spectral radius of a bounded linear operator A ∈ BL(X). Find the spectral radius of A in BL, where A is given by the matrix
with respect to the standard basis of .
Let X be a Banach space, Y be a normed linear space and be a subset of B(X, Y). If
is not uniformly bounded, then there exists a dense subset D of X such that for every
is not bounded in Y.
Let X = C00 with Give an example of a Cauchy sequence in X that do not converge in X. Justify your choice of example.
Read the proof of the closed graph theorem carefully and explain where and how we have used the following facts in the proof.
i) X is a Banach space.
ii) Y is a Banach space.
iii) F is a closed map.
iv) Which property of continuity is being established to conclude that F is continuous.
Let H=R³ and F be the set of all x = (x1, x2, x3) in H such that x1 = 0. Find F1. Verify that every x ∈ H can be expressed as x = y + z where y ∈ Fand z ∈ F1.
Let X be a Banach space and Y be a closed subspace of X. Let π: X → X/Y be canonical quotient map. Show that is open.