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The problem: Let M be the set of all [tex]f\in L^1 \left( \left[ 0,1\right] \right) [/tex] relative to the Lebesgue measure, such that

[tex]\int_{t=0}^{1}f(t)dt = 1[/tex].

Show that M is a closed convex subset of [tex]L^1 \left( \left[ 0,1\right] \right) [/tex] which contains infinitely many elements of minimal norm.

What I've got: Define the linear functional [tex]\Lambda f = \int_{t=0}^{1}f(t)dt[/tex]

Convexity is easy, for any f,g in M and s in (0,1) put [tex]h_s(t)=sf(t)+(1-s)h(t)[/tex] so that

[tex]\Lambda h_s = \Lambda \left( sf+(1-s)h\right) = s\Lambda f + (1-s)\Lambda g = s + (1-s) = 1\Rightarrow \int_{t=0}^{1}h_s(t)dt = 1\Rightarrow h_s\in M[/tex]

by which it is understood that M is convex.

That M is closed is my first trouble. Let [tex]\left\{ f_k\right\} \rightarrow f[/tex] be a sequence of vectors such that [tex]f_k\in M,\forall k\in\mathbb{N}[/tex]. Then we know that

[tex]\forall \epsilon >0, \exists N\in\mathbb{N}\mbox{ such that }k\geq N\Rightarrow \| f_k - f\| = \Lambda \left( |f_k - f|\right) < \epsilon[/tex]

How do I prove that [tex]f\in M[/tex] ? Can I show that [tex]\Lambda[/tex] is a bounded linear map (actually functional) from [tex]L^1 \left( \left[ 0,1\right] \right)[/tex] into [tex]\mathbb{C}[/tex]?

[tex]\int_{t=0}^{1}f(t)dt = 1[/tex].

Show that M is a closed convex subset of [tex]L^1 \left( \left[ 0,1\right] \right) [/tex] which contains infinitely many elements of minimal norm.

What I've got: Define the linear functional [tex]\Lambda f = \int_{t=0}^{1}f(t)dt[/tex]

Convexity is easy, for any f,g in M and s in (0,1) put [tex]h_s(t)=sf(t)+(1-s)h(t)[/tex] so that

[tex]\Lambda h_s = \Lambda \left( sf+(1-s)h\right) = s\Lambda f + (1-s)\Lambda g = s + (1-s) = 1\Rightarrow \int_{t=0}^{1}h_s(t)dt = 1\Rightarrow h_s\in M[/tex]

by which it is understood that M is convex.

That M is closed is my first trouble. Let [tex]\left\{ f_k\right\} \rightarrow f[/tex] be a sequence of vectors such that [tex]f_k\in M,\forall k\in\mathbb{N}[/tex]. Then we know that

[tex]\forall \epsilon >0, \exists N\in\mathbb{N}\mbox{ such that }k\geq N\Rightarrow \| f_k - f\| = \Lambda \left( |f_k - f|\right) < \epsilon[/tex]

How do I prove that [tex]f\in M[/tex] ? Can I show that [tex]\Lambda[/tex] is a bounded linear map (actually functional) from [tex]L^1 \left( \left[ 0,1\right] \right)[/tex] into [tex]\mathbb{C}[/tex]?

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