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I have this code:

g1[x_] := 1;
g2[x_] := x;
g3[x_] := x^2;
norm[r_, s_] := Integrate[r[x] s[x] + r'[x] s'[x], {x, 0, 1}];
gram = {
        {norm[g1, g1], norm[g1, g2], norm[g1, g3]},
        {norm[g2, g1], norm[g2, g2], norm[g2, g3]},
        {norm[g3, g1], norm[g3, g2], norm[g3, g3]}
       }

Which returns:

$$ \left( \begin{array}{ccc} 1 & \frac{1}{2} & \frac{1}{3} \\ \frac{1}{2} & \frac{4}{3} & \frac{5}{4} \\ \frac{1}{3} & \frac{5}{4} & \frac{23}{15} \\ \end{array} \right) $$

I wish I could do this computation using Map with just g[x_]:= {1,x,x^2} without loops, but I don't really understand the tricky Mathematica language for such things. I don't really know if I should use Apply or Map or something else.

I have been using Mathematica for a while and I've tried to learn this kind of coding without loops but I'm still like the first day. Watching the examples I saw here is not enough for me. I just can see a bunch of # /& @ without been able to understand it. So I always ended avoiding this chapter.

Anyone knows any tutorial for this particular matter?

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Here's one way...

g[1][x_] := 1;
g[2][x_] := x;
g[3][x_] := x^2;
norm[r_, s_] := Integrate[r[x]*s[x] + r'[x]*s'[x], {x, 0, 1}];

gram=Array[norm[g[#1], g[#2]] &, {3, 3}]

or...

gram=Table[norm[a, b], {a, {g1, g2, g3}}, {b, {g1, g2, g3}}];

or...

gram=Outer[norm, {g1, g2, g3}, {g1, g2, g3}]
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