Elementary Properties of Bounded Linear Maps and Functionals on Real Inner Product Spaces

Let $\mathbb{R}$ be the ordered field of real numbers, with the notation of that item, whose order is in particular a total order, and for $s\in\mathbb{R}$ let $|s|$ be its absolute value. Let $E$, $F$ and $G$ be real inner product spaces, with inner products, norms and distances in the ambient notation; each distance is a metric by The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity (iii). Let $\mathcal{L}(E,F)$ be the set of bounded linear maps from $E$ to $F$, with $\mathrm{id}_{E}$ the identity map of $E$ as in that clause, let $\lVert\cdot\rVert$ be the operator norm, let $0_{E}$ and $0_{F}$ be the zero vectors, and let $d_{\mathbb{R}}(s,t)=|s-t|$ be the metric on $\mathbb{R}$ of The Absolute Value Metric on the Real Numbers. For maps $S,T:E\to F$ and $\lambda\in\mathbb{R}$, $S+T$ and $\lambda T$ denote the maps $x\mapsto Sx+Tx$ and $x\mapsto\lambda\,Tx$. Then the following hold.

1. (Norm bound)(i) For $T\in\mathcal{L}(E,F)$ and $x\in E$, $|Tx|_{F}\le\lVert T\rVert\,|x|_{E}$; and if $C$ is a real number with $0\le C$ and $|Tx|_{F}\le C|x|_{E}$ for every $x\in E$, then $\lVert T\rVert\le C$.

2. (Supremum formula)(ii) For $T\in\mathcal{L}(E,F)$, $\lVert T\rVert$ is the least upper bound of the set $\{\,|Tx|_{F}\ :\ x\in E,\ |x|_{E}\le 1\,\}$.

3. (Boundedness, Lipschitz continuity and continuity)(iii) For a linear map $T:E\to F$ the following are equivalent: $T$ is bounded; $T$ is Lipschitz from $(E,d_{E})$ to $(F,d_{F})$; $T$ is continuous on $E$; $T$ is continuous at $0_{E}$ relative to $E$. When $T$ is bounded it is Lipschitz with constant $\lVert T\rVert$.

4. (Vector space and norm)(iv) If $S,T\in\mathcal{L}(E,F)$ and $\lambda\in\mathbb{R}$, then $S+T\in\mathcal{L}(E,F)$ and $\lambda T\in\mathcal{L}(E,F)$, with $\lVert S+T\rVert\le\lVert S\rVert+\lVert T\rVert$ and $\lVert\lambda T\rVert=|\lambda|\,\lVert T\rVert$; $\lVert T\rVert=0$ if and only if $Tx=0_{F}$ for every $x\in E$; and $\mathcal{L}(E,F)$ with these operations is a vector space over $\mathbb{R}$, with zero vector the map $x\mapsto 0_{F}$.

5. (Composition)(v) If $T\in\mathcal{L}(E,F)$ and $S\in\mathcal{L}(F,G)$, then the composition $S\circ T$ belongs to $\mathcal{L}(E,G)$ and $\lVert S\circ T\rVert\le\lVert S\rVert\,\lVert T\rVert$. Moreover $\mathrm{id}_{E}\in\mathcal{L}(E)$, and $\lVert\mathrm{id}_{E}\rVert=1$ whenever $E\ne\{0_{E}\}$.

6. (Kernel)(vi) For $T\in\mathcal{L}(E,F)$ the set $\ker T=\{x\in E: Tx=0_{F}\}$ is a closed linear subspace of $E$.

7. (The real line as an inner product space)(vii) The field $\mathbb{R}$, with its addition as vector addition and its multiplication as scalar multiplication, is a vector space over $\mathbb{R}$, and $\langle s,t\rangle=st$ is an inner product on it; the resulting real inner product space, again denoted $\mathbb{R}$, has norm $|s|$ (the absolute value) and distance $d_{\mathbb{R}}$. A linear functional on $E$ is the same thing as a linear map from $E$ to this space; it is bounded as a functional if and only if it is bounded as a linear map $E\to\mathbb{R}$, and then its norm as a functional equals its operator norm.

8. (Functionals)(viii) Claims 1, 2, 3 and 6 hold for bounded linear functionals $\ell$ on $E$, with $|\ell(x)|$ in place of $|Tx|_{F}$, $\lVert\ell\rVert$ in place of $\lVert T\rVert$, $(\mathbb{R},d_{\mathbb{R}})$ in place of $(F,d_{F})$ and $\ker\ell=\{x\in E:\ell(x)=0\}$. Moreover, for every $z\in E$ the map $x\mapsto\langle x,z\rangle_{E}$ is a bounded linear functional on $E$ with norm $|z|_{E}$.