If you want your classes without repetitions, then use the following XSLT:
<?xml version="1.0" encoding="UTF-8" ?>
<xsl:transform version="1.0"
xmlns:xsl="http://www.w3.org/1999/XSL/Transform"
xmlns:exsl="http://exslt.org/common"
extension-element-prefixes="exsl">
<xsl:output method="xml" encoding="UTF-8" indent="yes" />
<xsl:template match="/">
<document>
<xsl:variable name="cls1">
<xsl:for-each select=".//header/@class">
<xsl:call-template name="tokenize">
<xsl:with-param name="txt" select="."/>
</xsl:call-template>
</xsl:for-each>
</xsl:variable>
<xsl:variable name="cls" select="exsl:node-set($cls1)"/>
<header>
<xsl:attribute name="class">
<xsl:for-each select="$cls/*[not(preceding-sibling::* = .)]">
<xsl:value-of select="."/>
<xsl:if test="position() < last()">
<xsl:text> </xsl:text>
</xsl:if>
</xsl:for-each>
</xsl:attribute>
</header>
</document>
</xsl:template>
<xsl:template name="tokenize">
<xsl:param name="txt"/>
<xsl:if test="$txt">
<xsl:if test="contains($txt, ' ')">
<cls>
<xsl:value-of select="substring-before($txt, ' ')"/>
</cls>
<xsl:call-template name="tokenize">
<xsl:with-param name="txt" select="substring-after($txt, ' ')"/>
</xsl:call-template>
</xsl:if>
<xsl:if test="not(contains($txt, ' '))">
<cls>
<xsl:value-of select="$txt"/>
</cls>
</xsl:if>
</xsl:if>
</xsl:template>
</xsl:transform>
In XSLT 1.0 it is much more difficult than in XSLT 2.0, but as you see,
it is possible.
Edit
A revised version using Muenchian grouping (inspired by comment from Michael):
<?xml version="1.0" encoding="UTF-8" ?>
<xsl:transform version="1.0"
xmlns:xsl="http://www.w3.org/1999/XSL/Transform"
xmlns:exsl="http://exslt.org/common"
extension-element-prefixes="exsl">
<xsl:output method="xml" encoding="UTF-8" indent="yes" />
<xsl:variable name="cls1">
<xsl:for-each select=".//header/@class">
<xsl:call-template name="tokenize">
<xsl:with-param name="txt" select="."/>
</xsl:call-template>
</xsl:for-each>
</xsl:variable>
<xsl:variable name="cls2" select="exsl:node-set($cls1)"/>
<xsl:key name="classKey" match="cls" use="."/>
<xsl:template match="/">
<document>
<header>
<xsl:attribute name="class">
<xsl:for-each select="$cls2/*[generate-id() = generate-id(key('classKey', .)[1])]">
<xsl:value-of select="."/>
<xsl:if test="position() < last()">
<xsl:text> </xsl:text>
</xsl:if>
</xsl:for-each>
</xsl:attribute>
</header>
</document>
</xsl:template>
<xsl:template name="tokenize">
<xsl:param name="txt"/>
<xsl:if test="$txt">
<xsl:if test="contains($txt, ' ')">
<cls>
<xsl:value-of select="substring-before($txt, ' ')"/>
</cls>
<xsl:call-template name="tokenize">
<xsl:with-param name="txt" select="substring-after($txt, ' ')"/>
</xsl:call-template>
</xsl:if>
<xsl:if test="not(contains($txt, ' '))">
<cls>
<xsl:value-of select="$txt"/>
</cls>
</xsl:if>
</xsl:if>
</xsl:template>
</xsl:transform>