C++ friend template function: Function definition not found

Question

While trying to wrap my head around C++ templates I have run into a compiler warning I do not understand. In below vec.h Visual Studio 2019 will put a green squiggle under operator<<<T> in class vec with the warning "Function definition for operator<<<T> not found.

The code otherwise compiles and runs fine. In particular I do not understand how this case differs from operator+ which does not generate a warning.

#pragma once
#include <iostream>

template<typename T>
class vec;

template<typename T>
vec<T> operator+(const vec<T>& a, const vec<T>& b);

template<typename T>
std::ostream& operator<<(std::ostream& os, const vec<T>& v);

template<typename T>
class vec {
public:
    vec(T xx = 0, T yy = 0) : x(xx), y(yy) {}
    friend vec<T> operator+<T>(const vec<T>& a, const vec<T>& b);
    friend std::ostream& operator<<<T>(std::ostream& os, const vec<T>& v);
private:
    T x, y;
};

In vec.cpp I put the function templates for the friend functions and some explicit instantiations.

#include <iostream>
#include "vec.h"

template<typename T>
vec<T> operator+(const vec<T>& a, const vec<T>& b) {
    return vec<T>(a.x + b.x, a.y + b.y);
}

template vec<int> operator+(const vec<int>& a, const vec<int>& b);
template vec<float> operator+(const vec<float>& a, const vec<float>& b);

template<typename T>
std::ostream& operator<<(std::ostream& os, const vec<T>& v) {
    return os << "<" << v.x << "," << v.y << ">";
}

template std::ostream& operator<<(std::ostream& os, const vec<int>& v);
template std::ostream& operator<<(std::ostream& os, const vec<float>& v);

Answer 1

This is just an incorrect warning by Visual Studio IntelliSense. The code is well-formed and is compiled successfully.