dp=Function[{f,g},Integrate[f*g,{x,-1,1}]]

nm=Function[f,Sqrt[dp[f,f]]]

dp[x^2,x^4]

nm[x]

w[0]=1

v[0]=w[0]/nm[w[0]]

w[1]=x-dp[v[0],x]*v[0]

v[1]=w[1]/nm[w[1]]

w[2]=x^2-dp[v[0],x^2]*v[0]-dp[v[1],x^2]*v[1]

v[2]= Expand[w[2]/nm[w[2]]]

N0=8

For[k=0,k≤N0,k++,
w[k]=x^k;
For[j=0,j<k,j++,
w[k]=w[k]-dp[v[j],x^k]*v[j]];
v[k]=Simplify[w[k]/nm[w[k]]];
Print["v[",k,"]=",v[k]]]

Plot[v[5],{x,-1,1}]

R5=x/.Solve[v[5]==0,x]

R5n=N[R5,16]