The role of dual space of a normed space in functional analysisWe have known that dual space of a normed space is very important in functional analysis.I would like to ask two questions related dual space of a normed space:What is the motivation of constructing dual space of a normed space?What is the main role of dual space of a normed space in functional analysis?

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Yadira Mcdowell · Accepted Answer

When we consider finite dimensional spaces V, we do this usually by choosing a basis       v    1    ,  …  ,      v    n   and look at the hereby given isomorphism   T  :  V  →            K        n   with   T      v    i    =      e    i  . Another way to see this is that we are describing a general   v  ∈  V by its coordinates       λ    i    (  v  )  :=      π    i    (  T  v  ) where       π    i    :            K        n    →      K   is the projection onto the i-th factor. It&#039;s helpful to have a unique description by real numbers at hand, for we can use all properties of K we have already established. The main point here is that knowing all       λ    i    (  v  ) gives us v.In infinite dimensions, generally we have no basis to hand. AC gives us the existence, but this is not very helpful in computations. The idea of considering functionals may be seen as a generalization of the       λ    i    :  V  →      K   from above. Instead of looking at some (well choosen) functionals from V to K we look at all of them, that is, at the set       X    ∗    =  {      x    ∗    :  V  →      K    ∣      x    ∗     linear, continuous  } linear, continuous} and find that it works quite well as a replacement of the coordinate functionals.For example, Hahn-Banach helps us to distiguish elements of X in the following sense:  x  ≠  y    ⟺    ∃      x    ∗    :      x    ∗    (  x  )  ≠      x    ∗    (  y  )  .As with the coordinates, our generialized &quot;coordinates&quot; (x∗(x)) for   x  ∈  X allow us to reformulate problems in X to problems in K, which are often easier to solve.

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