I am studying some theories around FEM method in 2D, and I am trying to solve this problem from Ciarlet's book (the proof was not provided): Consider a simplex $T$ in $R^d$ with $N\_1(T) = \left\{N\_i\right\}\_{i=0}^{d}\subset P\_1^{\*}(T)$ be the Lagrange nodal variables (or nodal evaluation). By the Riesz representation theorem, there exist functions $\lambda\_{i}^{\*}\in P\_1(T)$ (which is the representation of $N\_i\in P\_1(T)$) such that $N\_j(\lambda\_{i}) = \int\_{T} \lambda\_{i}\lambda\_{j}^{\*} = \delta\_{ij}$ for each $0\leq i\leq d$. Show that $\lambda\_{i}^{\*} = \frac{(1+d)^2}{|T|}\lambda\_{i} - \frac{1+d}{|T|}\sum\_{j\neq i} \lambda\_j$


**My attempt:** I was thinking that $\lambda\_i^{\*}$ plays a role like a Fourier coefficient, so I multiply both sides by $\lambda\_j$ and integrate over $T$. Then LHS would be $\delta\_{ji}$, while RHS $= \frac{(1+d)^2}{|T|}\int\_{T} \lambda\_{i}\lambda\_{j} - \frac{1+d}{|T|}\sum\_{j\neq i}\int\_{T} \lambda\_{i}\lambda\_{j}$. Now, for $i=j$, then $LHS=1$, and I was thinking of $\int\_{T} \lambda\_{i}\lambda\_{i} = $ the area of region $T$, but then it would result in $(1+d)^2,$ which is certainly not equal to $1$ (contradiction). I got stuck here completely....


**My question:** Could anyone please help me with this problem? It seems to be such a beautiful identity, yet quite hard to prove. Any thoughts would really be appreciated.



Dual basis of Lagrange nodal variables in $R^d$
(failed to edit the comment-answer in 5 minutes fully)


In fact, no, I am sure there is a nice way to compute all this but now can think only of a brute force one. $\int\_{T^{hat}} \lambda\_1^2 = \int\_{T^{hat}} x\_1^2 dx = \int\_0^1 x\_1^2 V\_{d-1}(1-x\_1) dx\_1$ where $V\_k(r)$ is the volume of $k$-dimensional canonical simplex with side lentghs equal to $r$. So, I guess $V\_k(r) = r^k V\_k(1) = r^k / k!$. So I got something like $\int\_{T^{hat}} \lambda\_1^2 dx = 1/(d-1)! \cdot 2 / (d(d+1)(d+2)) = \frac{2}{ (d+2)!}$. Which differs from your value.
The integral for a product $\lambda\_i \lambda\_j$ is more tedious, since leads to something like 2d-integral $\int\_0^1 x\_1 \int\_0^1 x\_2 (1-x\_1-x\_2)^{d-2} dx\_1 dx\_2 \cdot \frac{1}{(d-2)!}$ which is doable.
Hope someone would come up with a nice and easy answer to overall question.


