# Question about proof of Hahn-Banach lemma

I think they do something unnecessary in my notes in the proof of the following lemma:

The idea of the proof is to partially order the set $\Sigma$ of pairs $(X_i, f_i)$ where $X_i$ is a linear subspace of $X$ containing $Y$ and $f_i : X_i \to \mathbb R$ is a linear map. Then apply Zorn to get a maximal element $(X^\prime, f^\prime)$ and show that $X^\prime = X$. The way to do this is by contradiction. One assumes that there is a point $z$ in $X \setminus X^\prime$ and then defines a $g$ such that $(X^{\prime \prime}, g)$ where $X^{\prime \prime}$ is spanned by $z$ and $X^\prime$ is strictly greater than $(X^\prime, f^\prime)$ in $\Sigma$, contradicting maximality.

In the notes we define $g$ in terms of $f^\prime$: If $z \in X \setminus X^\prime$ and $y \in X^{\prime \prime}$ then $g(y) = f^\prime(x^\prime) + \lambda c$ for some $\lambda$ and some $x^\prime \in X^\prime$. All we need to show is $g \leq p$. And this is the part where I think they are doing something superfluous. They pick $c \in [A,B]$ (see below) where I think it's enough to pick $c \leq B$:

If those pngs aren't readable then you can also find the proof in these notes at the very start of chapter 6. Thanks for your help.

-

3
How do you prove (6.2) and treat the case of $\lambda \lt 0$ in (6.3) if $c \lt A$? – t.b. Aug 14 '12 at 10:13
@t.b. Let me see, I think I ignored the case $\lambda < 0$ in my proof! : ) Thank you! – Matt N. Aug 14 '12 at 10:14

1
In your sketch of the Zornication part of the proof you probably wanted "$f$ is a linear function majorized by $p$" instead of "$f$ is a continuous linear functional". – Martin Sleziak Aug 14 '12 at 10:55
@MartinSleziak Yep, that's right! Thank you! – Matt N. Aug 14 '12 at 10:59
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 33, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9390620589256287, "perplexity_flag": "head"}