In the proof of Lemma 4.1, pp. 962-963 in "Endpoint Strichartz Estimates" by Tao and Keel (1997) (see MR1646048 or Zbl 0922.35028), the authors first proved the statements hold for some boundary and vertex values of (a,b), and then it was claimed that the statements of the Lemma hold true for points in some neighborhood of $(r,r)$ since $(r,r)$ is in the convex hull determined by those boundary and vertex values of $(a,b)$. This is due to interpolation for bilinear forms, according to the expression in the article.
 Question. I wonder what is interpolation for bilinear forms and how it is used here. Is there some books including this? I merely heard of interpolation for operators between function spaces, and interpolation for functionals of single variable as well. But never saw interpolation for bilinear forms before...

In the proof of Lemma 4.1, pp. 962–963 in "Endpoint Strichartz Estimates" by Tao and Keel (1997) (see MR1646048 or Zbl 0922.35028), the authors first proved the statements hold for some boundary and vertex values of $(a,b)$, and then it was claimed that the statements of the Lemma hold true for points in some neighborhood of $(r,r)$ since $(r,r)$ is in the convex hull determined by those boundary and vertex values of $(a,b)$. This is due to interpolation for bilinear forms, according to the expression in the article. 

Question. I wonder what is interpolation for bilinear forms and how it is used here. Is there some books including this? I merely heard of interpolation for operators between function spaces, and interpolation for functionals of single variable as well. But never saw interpolation for bilinear forms before...

In the proof of Lemma 4.1 in Endpoint Strichartz Estimates by Tao in 1997, the author first proved the statements hold for some boundary and vertex values of (a,b), and then it was claimed that the statements of the Lemma hold true for points in some neighborhood of (r,r) since (r,r) is in the convex hull determined by those boundary and vertex values of (a,b). This is by the interpolation for bilinear forms according to the expression in the artical. I wonder what is interpolation for bilinear forms and how it is used here? Is there some books including this? I merely heard of interpolation for operators between function spaces, and interpolation for functionals of single variable as well. But never saw interpolation for bilinear forms before...

In the proof of Lemma 4.1, pp. 962-963 in "Endpoint Strichartz Estimates" by Tao and Keel (1997) (see MR1646048 or Zbl 0922.35028), the authors first proved the statements hold for some boundary and vertex values of (a,b), and then it was claimed that the statements of the Lemma hold true for points in some neighborhood of $(r,r)$ since $(r,r)$ is in the convex hull determined by those boundary and vertex values of $(a,b)$. This is due to interpolation for bilinear forms, according to the expression in the article.
Question. I wonder what is interpolation for bilinear forms and how it is used here. Is there some books including this? I merely heard of interpolation for operators between function spaces, and interpolation for functionals of single variable as well. But never saw interpolation for bilinear forms before...

In the proof of Lemma 4.1 in Endpoint Strichartz Estimates by Tao in 1997, the author first proved the statements hold for some boundary and vertex values of (a,b), and then it was claimed that the statements of the Lemma hold true for points in some neighborhood of (r,r) since (r,r) is in the convex hull determined by those boundary and vertex values of (a,b). This is by the interpolation for bilinear forms according to the expression in the artical. I wonder what is interpolation for bilinear forms and how it is used here? Is there some books including this? I merely heard of interpolation for operators between function spaces, and interpolation for functionals of single variable as well. But never saw interpolation for bilinear forms before...

The lemma in question is shown in the following srceenshot.

In the proof of Lemma 4.1 in Endpoint Strichartz Estimates by Tao in 1997, the author first proved the statements hold for some boundary and vertex values of (a,b), and then it was claimed that the statements of the Lemma hold true for points in some neighborhood of (r,r) since (r,r) is in the convex hull determined by those boundary and vertex values of (a,b). This is by the interpolation for bilinear forms according to the expression in the artical. I wonder what is interpolation for bilinear forms and how it is used here? Is there some books including this? I merely heard of interpolation for operators between function spaces, and interpolation for functionals of single variable as well. But never saw interpolation for bilinear forms before...