Math Problems Vault

Problem 2.1

Prove that a scalar product in a Hilbert space is a continuous function in each of the arguments.

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Problem 2.2

(1)(2)

Let $L$ and $M$ be linear sets in a Hilbert space $H$ , and $L \perp M$ .

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(1)

Prove that for all $x \in L \bigoplus M$ the representation $x = x^{\prime }+x^{\prime \prime }, x^{\prime } \in L, x^{\prime \prime } \in M$ , is unique.

(2)

Let $H = L \bigoplus M$ . Prove that $M^{\perp } = L$ .

Problem 2.3

Find the orthogonal complement in $L_{2}(\left[-1,1\right])$ of the set of all even functions.

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Remark.

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A function $x \in L_{2}(\left[-1,1\right])$ is said to be even if $x(t) = x(-t)$ for almost all $t \in [-1,1]$ . The notion of an odd function is introduced in a similar way.

Problem 2.4

Let $L$ be a total set in a Hilbert space $H$ . Prove that $L^{\perp } = \left\{ 0\right\}$ .

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Problem 2.5

Find the orthogonal complement of the set $M = \left\{ e^{k t} \mid k \geq 10\right\}$ in $L_{2}(\left[0,1\right])$ .

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