Bounded point evaluations force finite dimension

[3/33%]

Show

LaTeX

Show LaTeX

Problem V.4.7 from Conway J.B. "A Course in Functional Analysis" [Con07]More problems in Section V.4 "Reflexivity Revisited"

Let $X$ be compact and suppose there is a norm on $C(X)$ that is given by an inner product making $C(X)$ into a Hilbert space such that for every $x$ in $X$ the functional $f \mapsto f(x)$ on $C(X)$ is continuous with respect to the Hilbert space norm. Show that $X$ is finite.

Problem 12.3.49 from Petrovic J.S. "Measure and Integral: Theory and Practice" [Pet25]More problems in Section 12.3 "The Dual of $L^{p}$"

Let $S$ be a subspace of $C[0,1]$ and suppose that there exists a constant $K \geq 0$ such that $\left\| f\right\|_{\max } \leq K\left\| f\right\|_{2}$ for all $f \in S$ . Prove that $S$ is finitedimensional.

Problem 12.3.51 from Petrovic J.S. "Measure and Integral: Theory and Practice" [Pet25]More problems in Section 12.3 "The Dual of $L^{p}$"

(a)(b)(c)

Let $X$ be a closed subspace of $L^{2}\left(0,1\right)$ which is contained in $C[0,1]$ . Prove that:

(a)

There exists $A>0$ such that $\left\| f\right\|_{\max } \leq A\left\| f\right\|_{2}$ for all $f \in X$ .

(b)

For each $x \in [0,1]$ there exists $h_{x} \in X$ such that $f(x)= \left\langle f, h_{x} \right\rangle$ for all $f \in X$ , and $\left\| h_{x}\right\|_{2} \leq A$ .

(c)

The dimension of $X$ is at most $A^{2}$ .

References

[Con07] Conway J.B. A Course in Functional Analysis. 2nd ed., Springer, 2007.

[Pet25] Petrovic J.S. Measure and Integral: Theory and Practice. CRC Press, 2025.