Hilbert Space

Prove that $\left\langle x, y \right\rangle =\sum_{k=1}^{\infty } x_{k} y_{k}$ is an inner product on $\ell^{2}$.

Prove that $\left\langle f, g \right\rangle =\int_{X} f g d \mu$ is an inner product on $L^{2}\left(X, \mu \right)$.

Prove Proposition 10.5.4. Use it to derive Corollary 10.5.6.

Proposition 10.5.4 (Cauchy-Schwarz Inequality): For any $x, y \in H$, $\left| \left\langle x, y \right\rangle \right| \leq \left\| x\right\| \left\| y\right\|$.

Corollary 10.5.6: For a fixed $y \in H$ the inner product $\left\langle x, y \right\rangle$ is a continuous function of $x$ in the norm topology. An analogous statement is true when we fix $x$.

Prove that equality holds in Cauchy-Schwarz inequality if and only if $x=0$ or $y=\lambda x$, for some $\lambda \in \mathbb {R}$.

Prove Proposition 10.5.5.

Proposition 10.5.5: The function $\left\| \cdot \right\|$ defined by $\left\| h\right\| = \sqrt{ \left\langle h, h \right\rangle }$ (formula (10.13)) is a norm on $H$.

Prove Proposition 10.5.7.

Proposition 10.5.7 (The Parallelogram Identity): For any $x, y \in H$, $\left\| x-y\right\|^{2} + \left\| x+y\right\|^{2} = 2\left\| x\right\|^{2} + 2\left\| y\right\|^{2}$.

Prove that the sequence $\left\{ f_{n}\right\}$ in Example 10.5.13 is Cauchy but not convergent.

Example 10.5.13 (an inner product space that is not a Hilbert space): Let $H$ be the set $C[-1,1]$ with the inner product $\left\langle f, g \right\rangle = \int_{-1}^{1} fg$. Define $f_{n}(x) = 0$ for $-1 \leq x < 0$, $f_{n}(x) = nx$ for $0 \leq x < 1/n$, and $f_{n}(x) = 1$ for $1/n \leq x \leq 1$.

Prove that $L^{p}\left(0,1\right)$ is not Hilbert space if $p \neq 2$.

Prove that $\ell^{p}$ is not Hilbert space if $p \neq 2$.

Prove that $C[a, b]$ is not Hilbert space.

Let $X$ be a normed linear space in which the Parallelogram Identity holds. Prove that there is an inner product on $X$ such that $\left\| x\right\|^{2}= \left\langle x, x \right\rangle$, for all $\in X$.

Let $\mathcal{H}$ be the set of all absolutely continuous functions $f:[0,1] \rightarrow \mathbb {R}$ such that $f(0)=0$ and $f^{\prime } \in L^{2}\left(0,1\right)$. Let $\left\langle f, g \right\rangle =\int_{0}^{1} f^{\prime }(t) g^{\prime }(t) d t$ for $f, g \in \mathcal{H}$. Prove that $\mathcal{H}$ is Hilbert space.

Prove the Polarization Identity: For any $x, y \in \mathcal{H}$,

A vector $h$ in Hilbert space $\mathcal{H}$ is an extreme point of a set $A$ if

Prove that every unit vector in $\mathcal{H}$ is an extreme point of the unit sphere.

Let $\left\{ x_{n}\right\} ,\left\{ y_{n}\right\} , x, y$ be vectors in Hilbert space such that $\lim x_{n}=x$ and $\lim y_{n}=y$. Prove that $\lim \left\langle x_{n}, y_{n} \right\rangle = \left\langle x, y \right\rangle$.

Let ( $X, \mathfrak {M}, \mu$ ) be a measure space and let $\left\{ f_{n}\right\} , f \in L^{2}\left(X, \mu \right)$ such that

Prove that $\lim f_{n}=f$ in the norm of $L^{2}\left(X, \mu \right)$.

Let $\mathcal{H}$ be Hilbert space and let $X$ denote the set of all sequences $\mathbf{x}=\left\{ x_{n}\right\}$, where for each $n \in \mathbb {N}, x_{n} \in \mathcal{H}$ and $\left\| \mathbf{x}\right\| :=\sup \left\{ \left\| x_{n}\right\| : n \in \mathbb {N}\right\} <\infty$. Prove that with the natural definition of addition and scalar multiplication $X$ is a normed linear space. Is it a Banach space? Is it Hilbert space?

In Exercises 10.5.18-10.5.21 a vector subspace $V$ of $L^{2}\left(0,1\right)$ is given consisting of some polynomials $p(x)=\sum_{k=0}^{\operatorname {deg} p} c_{k} x^{k}$. Find its orthogonal complement.

All polynomials.

In Exercises 10.5.18-10.5.21 a vector subspace $V$ of $L^{2}\left(0,1\right)$ is given consisting of some polynomials $p(x)=\sum_{k=0}^{\operatorname {deg} p} c_{k} x^{k}$. Find its orthogonal complement.

Polynomials of $x^{2}$, i.e., such that $c_{2 k-1}=0$, for all $k \in \mathbb {N}$.

In Exercises 10.5.18-10.5.21 a vector subspace $V$ of $L^{2}\left(0,1\right)$ is given consisting of some polynomials $p(x)=\sum_{k=0}^{\operatorname {deg} p} c_{k} x^{k}$. Find its orthogonal complement.

Polynomials with the property that $c_{0}=0$.

In Exercises 10.5.18-10.5.21 a vector subspace $V$ of $L^{2}\left(0,1\right)$ is given consisting of some polynomials $p(x)=\sum_{k=0}^{\operatorname {deg} p} c_{k} x^{k}$. Find its orthogonal complement.

Polynomials satisfying $\sum_{k=0}^{\operatorname {deg} p} c_{k}=0$.

Let $\mathcal{K}$ be a subspace of Hilbert space $\mathcal{H}$. Prove that $\mathcal{K} \cap \mathcal{K}^{\perp }$ consists of zero vector alone.

Let $\mathcal{K}, \mathcal{M}$ be two subspaces of Hilbert space $\mathcal{H}$ such that $\mathcal{K} \subset \mathcal{M}$. Prove that $\mathcal{M}^{\perp } \subset \mathcal{K}^{\perp }$.

Let $S$ be a subset of Hilbert space. Prove that $\left(S^{\perp }\right)^{\perp }$ equals the closed linear span of vectors in $S$.

Let $V$ be a closed subspace of Hilbert space $\mathcal{H}$, and let $x_{0} \in \mathcal{H}$. Show that

Let $S$ be a subset of $\mathcal{H}$. Prove that $S^{\perp }=S^{\perp \perp \perp }$.

Let $\mathcal{K}, \mathcal{M}$ be two closed subspaces of Hilbert space $\mathcal{H}$ that are orthogonal to each other. Prove that $\mathcal{K}+\mathcal{M}$ is a closed subspace of $\mathcal{H}$.

In Exercises 10.5.28-10.5.30 $X$ is the vector space of all finite sequences equipped with the $\ell^{2}$ norm and the inner product. For the given subspace $\mathcal{M} \subset X$ determine $\overline{\mathcal{M}}$ and $\mathcal{M}^{\perp }$.

$\mathcal{M}=\left\{ x=\left\{ x_{n}\right\} \in X: \sum_{n} x_{n}=0\right\}$.

In Exercises 10.5.28-10.5.30 $X$ is the vector space of all finite sequences equipped with the $\ell^{2}$ norm and the inner product. For the given subspace $\mathcal{M} \subset X$ determine $\overline{\mathcal{M}}$ and $\mathcal{M}^{\perp }$.

$\mathcal{M}=\left\{ x=\left\{ x_{n}\right\} \in X: \sum_{n} x_{n} / n=0\right\}$.

In Exercises 10.5.28-10.5.30 $X$ is the vector space of all finite sequences equipped with the $\ell^{2}$ norm and the inner product. For the given subspace $\mathcal{M} \subset X$ determine $\overline{\mathcal{M}}$ and $\mathcal{M}^{\perp }$.

$\mathcal{M}=\left\{ x=\left\{ x_{n}\right\} \in X: \sum_{n} x_{n} / \sqrt{n}=0\right\}$.

Let $\mathcal{K}, \mathcal{M}$ be two closed subspaces of Hilbert space $\mathcal{H}$. Prove that

Let $\mathcal{M}$ be a subspace of Hilbert space $\mathcal{H}$, let $x \in \mathcal{H} \backslash \mathcal{M}$, let $d= d(x, \mathcal{M})$, and let $\left\{ x_{n}\right\} \subset \mathcal{M}$ be such that $\lim_{n}\left\| x_{n}-x\right\| =d$. Prove that $\left\{ x_{n}\right\}$ is a Cauchy sequence in $\mathcal{M}$.