Section 3.1

Examples of Problems with Solutions

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Problem 3.1

Determine whether the given functionals are linear and continuous. For continuous linear functionals, find their norms using the definition.

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(1)

X=C([a,b]),f(x)=x(a+b2)X = C(\left[a, b\right]), f(x) = x\left(\frac{a+b}{2}\right);

(2)

X=C([a,b]),f(x)=x(a)−x(b)X = C(\left[a, b\right]), f(x) = x(a)-x(b);

(3)

X=C([−1,1]),f(x)=∫−11x(t)sign⁡tdtX = C(\left[-1,1\right]), f(x) = \int_{-1}^{1} x(t) \operatorname {sign} t d t;

(4)

X=C([0,1]),f(x)=∫01t∣x(t)∣dt;X = C(\left[0,1\right]), f(x) = \int_{0}^{1} t\left|x(t)\right| d t ;

(5)

X=C([0,1])X = C(\left[0,1\right]) with the norm ∥x∥=∫01∣x(t)∣dt,f(x)=x(1)\left\| x\right\| = \int_{0}^{1}\left|x(t)\right| d t, f(x) = x(1);

(6)

X=Lp([−1,1]),1≤p≤+∞,f(x)=∫−11t2x(t)sign⁡tdtX = L_{p}(\left[-1,1\right]), 1 \leq p \leq +\infty , f(x) = \int_{-1}^{1} t^{2} x(t) \operatorname {sign} t d t;

(7)

X=l2,f(x)=∑n=1∞12n(xn+xn+1)X = l_{2}, f(x) = \sum_{n = 1}^{\infty } \frac{1}{2^{n}}\left(x_{n}+x_{n+1}\right);

(8)

X={x∈l2∣∑n=1∞xncos⁡n is a convergent series},∥⋅∥X=∥⋅∥l2X = \left\{ x \in l_{2} \mid \sum_{n = 1}^{\infty } x_{n} \cos n \text{ is a convergent series}\right\} , \left\| \cdot \right\|_{X} = \left\| \cdot \right\|_{l_{2}},

f(x)=∑n=1∞xncos⁡n,x∈X f(x) = \sum _{n = 1}^{\infty } x_{n} \cos n, x \in X
Problem 3.2

Let XX be a Banach space, ff be a linear functional on XX. Prove that

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(1)

Ker ff is a linear set, moreover if f∈X∗f \in X^{*} then Ker⁡f\operatorname {Ker} f is a subspace of XX;

(2)

if z∈X\Ker⁡fz \in X \backslash \operatorname {Ker} f then

∀x∈X∃!y∈Ker⁡f∃!λ∈K:x=y+λz. \forall x \in X \exists !y \in \operatorname {Ker} f \exists !\lambda \in \mathbb {K}: x = y+\lambda z.
Problem 3.3

Using the description of the respective dual spaces, prove the linearity and continuity of the given functionals. Find the norms of continuous linear functionals.

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(1)

X=C([−1,1]),f(x)=∫−11tx(t)dt−x(0)X = C(\left[-1,1\right]), f(x) = \int_{-1}^{1} t x(t) d t-x(0);

(2)

X=C([a,b]),f(x)=x(a+b2)X = C(\left[a, b\right]), f(x) = x\left(\frac{a+b}{2}\right);

(3)

X=C([0,3]),f(x)=∫01x(t)dt+∫23tx(t)dtX = C(\left[0,3\right]), f(x) = \int_{0}^{1} x(t) d t+\int_{2}^{3} t x(t) d t;

(4)

X=l2,f(x)=∑k=1∞12k(xk+xk+1)X = l_{2}, f(x) = \sum_{k = 1}^{\infty } \frac{1}{2^{k}}\left(x_{k}+x_{k+1}\right);

(5)

X=lp,1≤p≤+∞,f(x)=xj+xj+1,jX = l_{p}, 1 \leq p \leq +\infty , f(x) = x_{j}+x_{j+1}, j is fixed;

(6)

X=lp,1≤p≤+∞,f(x)=∑k=1∞xk3kX = l_{p}, 1 \leq p \leq +\infty , f(x) = \sum_{k = 1}^{\infty } \frac{x_{k}}{3^{k}};

(7)

X=Lp([0,1]),f(x)=∫012x(t)dt−∫231x(t)dtX = L_{p}(\left[0,1\right]), f(x) = \int_{0}^{\frac{1}{2}} x(t) d t-\int_{\frac{2}{3}}^{1} x(t) d t.

Problem 3.4

Let 1≤p<+∞1 \leq p < +\infty be fixed. For which α∈R\alpha \in \mathbb {R} do the functionals belong to the respective dual spaces? Calculate the norms of continuous linear functionals.

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(1)

X=Lp([1,+∞)),f(x)=∫1+∞x(t)tαdtX = L_{p}([1,+\infty )), f(x) = \int_{1}^{+\infty } \frac{x(t)}{t^{\alpha }} d t;

(2)

X=lp,f(x)=∑n=1∞αnxnX = l_{p}, f(x) = \sum_{n = 1}^{\infty } \alpha^{n} x_{n}.