Closed Graph, Banach Spaces and Continuous Linear Transformations

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  Solution: Recall: [ The space of all continuous functions f : [a, b] → K defined on a closed interval [a, b] becomes a Banach space if we define the norm of such a function as ||f|| = sup { |f(x)| : x in [a, b] }.

• x** is weak* continuous iff x** Є j(ε).

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• Banach Spaces

  109423 Banach Spaces Show that (X, ||*||) is a Banach space if and only if {x in X: ||x||=1} is complete. Know that in the first direction, we must show that {x in X: ||x||=1} is closed subset of X.

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  If A : V ---> W is a linear map, then show that it is continuous at every point v E V if and only if it is continuous at 0 E V. 3) Suppose A: (V, | * |) ---> (W, | * |w) is a linear map between normed spaces, and there is a number R E R such that |

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  76568 Banach Space and Closed Subspace Let I = [a,b] be a finite interval. Show that the space C(I,R^n) of continuous functions from I into R^n is a Banach space with the uniform norm llull = sup l u(t) l where t is in I.

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  594729 Linear Transformations on Finite Dimension Vector Spaces 2.

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  Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations.