Math Problems Vault

Problem 1

Prove that any ball in a normed space $X$ is convex.

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Problem 2

Consider the linear space $C[0,1]$ equipped with the norm

\left\| f\right\| _{1} = \int _{0}^{1}\left|f(x)\right| d x.

Prove that there is no inner product on $C[0,1]$ agreed with this norm.

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Problem 3

Consider the linear space $C[0,1]$ equipped with the norm

\left\| f\right\| = \max _{t \in [0,1]}\left|f(t)\right|.

Prove that there is no inner product on $C[0,1]$ agreed with this norm.

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Problem 4

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Problem 5

Prove that two any norms in a finite dimensional space $X$ are equivalent.

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Problem 6

(a)(b)

Let $X$ be a normed space.

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(a)

Find all subspaces of $X$ which are contained in some ball $B(a ; r)$ of $X$ .

(b)

Find all subspaces of $X$ which contain some ball $B\left(x_{0} ; \rho \right)$ of $X$ .

Problem 7

(a)(b)

Prove that any finite dimensional normed space :

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(a)

is complete (a Banach space),

(b)

is reflexive.

Problem 8

(a)(b)

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(a)

Prove that the inner product in a normed space $X$ can be recovered from the polarization identity:

\langle x, y\rangle = \frac{1}{4}\left[\left(\left\| x+y\right\| ^{2}-\left\| x-y\right\| ^{2}\right)-i\left(\left\| x+i y\right\| ^{2}-\left\| x-i y\right\| ^{2}\right)\right].

(b)

Prove that a normed space is an inner product space if and only if the norm satisfies the parallelogram law:

\left\| x+y\right\| ^{2}+\left\| x-y\right\| ^{2} = 2\left(\left\| x\right\| ^{2}+\left\| y\right\| ^{2}\right)

Problem 9

Let $X$ be an inner product space and let $\left\{ x_{1}, \ldots , x_{N}\right\}$ be an orthonormal set. Prove that

\left\| x-\sum _{n = 1}^{N} c_{n} x_{n}\right\|

is minimized by choosing $c_{n} = \left\langle x_{n}, x \right\rangle$ .

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Problem 10

Let $X$ be a normed space and $M$ a closed subspace of $X$ . Let $\pi : X \rightarrow X / M$ be the canonical map. Show that the topology induced by the standard norm on $X / M$ is the usual quotient topology, i.e. that $O \subset X / M$ is open in $X / M$ if and only if $\pi^{-1}(O)$ is open in $X$ .

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Problem 11

Let $X = C[0,1], M = \left\{ f \in C[0,1]: f(0) = 0\right\}$ . Show that $X / M = \mathbb {C}$ .

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Problem 12

If $0 < p < 1$ then $\ell^{p}$ is a vector space but $\left\| x\right\|_{p} = \left(\sum_{n}\left|x_{n}\right|^{p}\right)^{1 / p}$ is not a norm for $\ell^{p}$ .

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Problem 13

Suppose that $X$ is a linear space with inner product $\langle .,$ . $\rangle . If x_{n} \rightarrow x$ and $y_{n} \rightarrow y$ as $n \rightarrow \infty$ , prove that

\left\langle x_{n}, y_{n} \right\rangle \rightarrow \langle x, y\rangle \quad \text{ as } \quad n \rightarrow \infty .

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Problem 14

Prove that if $M$ is a closed subspace and $N$ is a finite dimensional subspace of a normed space $X$ , then $M+N: = \left\{ m+n: m \in M, n \in N\right\}$ is closed.

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