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Theorem 7. $X$ is a normed linear space over $\mathbb{C}, Y$ a linear subspace of $X$. For any $z$ in $X$, denote by $m(z)$ its distance from $Y$ : $$ m(z)=\inf _{y \in Y}|z-y| $$

We claim that for every $z$ in $X$ $$ m(z)=M(z),\tag{10} $$ where $$ M(z)=\max _{|l| \leq 1, l=0 \text { on } Y}|l(z)| .\tag{11} $$

Proof. Since the functionals $l$ entering the maximum problem (11) vanish on $Y$, and since $|l| \leq 1,|l(z)|=|l(z-y)| \leq|z-y|$ holds for all $y$ in $Y$; therefore $$ |l(z)| \leq \inf _{y \text { in } Y}|z-y|=m(z) . $$

It follows from this and the definition (11) of $M(z)$ that $$ M(z) \leq m(z) .\tag{12} $$

To show equality, we look at the linear space $Y_0$ consisting of all vectors of the form $y+a z, y$ in $Y, a$ complex, and define on $Y_0$ the linear functional $l_0$ : $$ l_0(y+a z)=a m(z) .\tag{13} $$

By definition (9) of $m$, it follows that $l_0$ is bounded on $Y_0$ by 1 ; so by theorem 4 , it can be extended to all of $X$ so that $\left|l_0\right|=1$. Set $y=0, a=1$ in (13): $$ l_0(z)=m(z) . $$

Combined with (12) this shows that $\ell_0$ solves the maximum problem (11), and that (10) holds.

MY Question is why we need over $\mathbb{C}$ in the statement of the theorem. I think it can be over $\mathbb{R}$.

One further question What is the fundamental difference between "over $\mathbb{R}$" and "over $\mathbb{C}$"?

evenzhou
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1 Answers1

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As a small remark, you should emphasise that $Y$ is a proper non-dense subspace of $X$ in the case you are dealing with in the proof.

For your first question, the same proof works if the vector space is over the real numbers.

For your second question, essentially all basic results in functional analysis hold in both the real and complex setting, with occasional obvious modifications when required.

Another useful point of view is for a vector space $X_{\mathbb{C}}$ over the complex numbers, you can also consider it as a vector space $X_{\mathbb{R}}$ over the real numbers. Every linear functional on $X_{\mathbb{C}}$ is a linear functional on $X_{\mathbb{R}}$, but the converse is not true. However, there is a one-to-one correspondence between the real part of a linear functional on $X_{\mathbb{C}}$ and a linear functional on $X_{\mathbb{R}}$. In other words, the real part of a complex-linear functional determines the whole linear functional.

Dean Miller
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