Functional analysis and Linear algebra problem!Problem is in the attached picture.Show that Real valued continuous functions on [a,b] are vector space, but in Rn Space. please check the attached image.I know that we have to prove the 10 axioms for vector space but i don't know how to apply this axioms here on Rn Space. Consider [a,b] c R, we define the set:C([a,b],R") = {f : [a,b] → R", f=(f,..f)/ f is continous on [a,b]}Show that the following statement is true:C([a,b],R") forms a linear space in relation to the estimation operations of two functions in C([a,b],R"),respectively of multiplication with real scalars with functions from C([a,b],R").

Consider the provided question, Let a,b⊂ℝ, we define the set: C([a,b],Rn)={ f : [a,b] →ℝn,…

Answered: Consider [a,b] c R, we define the set:…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Functional analysis and Linear algebra problem!

Problem is in the attached picture.

Show that Real valued continuous functions on [a,b] are vector space, but in Rⁿ Space. please check the attached image.

I know that we have to prove the 10 axioms for vector space but i don't know how to apply this axioms here on Rⁿ Space.

Consider [a,b] c R, we define the set:
C([a,b],R") = {f : [a,b] → R", f=(f,..f)/ f is continous on [a,b]}
Show that the following statement is true:
C([a,b],R") forms a linear space in relation to the estimation operations of two functions in C([a,b],R"),
respectively of multiplication with real scalars with functions from C([a,b],R").

Branch of mathematics concerned with mathematical structures that are closed under operations like addition and scalar multiplication. It is the study of linear combinations, vector spaces, lines and planes, and some mappings that are used to perform linear transformations. Linear algebra also includes vectors, matrices, and linear functions. It has many applications from mathematical physics to modern algebra and coding theory.

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