document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1184010>Question 1184010</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Show that C[a, b], together with the usual scalar multiplication and addition of functions, satisfies the eight axioms of a vector space </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #814597 by </B><A HREF=/tutors/aboutme.mpl?userid=robertb><B>robertb(5830)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=robertb><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=robertb><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=814597\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> \r<BR>\n" );
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document.write( "Let C[a,b] = the set of real-valued continuous functions over the interval [a,b].\r<BR>\n" );
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document.write( "Suppose f, g, and h are continuous functions over [a,b].\r<BR>\n" );
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document.write( "Since f + (g + h) = (f + g)+ h over [a,b], addition of functions is ASSOCIATIVE.\r<BR>\n" );
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document.write( "Since f + g = g + f  over [a,b], addition of functions is COMMUTATIVE.\r<BR>\n" );
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document.write( "Since the zero function 0 is continuous over [a,b] and 0 + f = f for any f in C[a,b], 0 is the IDENTITY element for C[a,b].\r<BR>\n" );
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document.write( "Since the function -f is also continuous over [a,b] and -f + f = 0, -f is the INVERSE element for any f that is in C[a,b].\r<BR>\n" );
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document.write( "Also, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=alpha%28beta%2Af%29+=+%28alpha%2Abeta%29f ALIGN=MIDDLE  ALT=\"alpha%28beta%2Af%29+=+%28alpha%2Abeta%29f\"   DISABLED_style= \"max-width:90%; height:auto;\" > for any f in C[a,b] and any real constants <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=alpha ALIGN=MIDDLE  ALT=\"alpha\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=beta ALIGN=MIDDLE  ALT=\"beta\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Af+=+f ALIGN=MIDDLE  ALT=\"1%2Af+=+f\"   DISABLED_style= \"max-width:90%; height:auto;\" >  for any f in C[a,b].\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=alpha%28f+%2B+g%29+=+alpha%2Af+%2B+alpha%2Ag ALIGN=MIDDLE  ALT=\"alpha%28f+%2B+g%29+=+alpha%2Af+%2B+alpha%2Ag\"   DISABLED_style= \"max-width:90%; height:auto;\" > for any f, g in C[a,b].\r<BR>\n" );
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document.write( "Finally, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28alpha+%2B+beta%29%2Af+=+alpha%2Af+%2B+beta%2Af ALIGN=MIDDLE  ALT=\"%28alpha+%2B+beta%29%2Af+=+alpha%2Af+%2B+beta%2Af\"   DISABLED_style= \"max-width:90%; height:auto;\" > for all f in C[a,b].\r<BR>\n" );
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document.write( "Therefore C[a,b] is a vector space with the usual scalar multiplication and addition of functions.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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