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<?php
function ajax_check_user_logged_in() {
    echo is_user_logged_in()?'yes':'no';
    die();
}
add_action('wp_ajax_is_user_logged_in', 'ajax_check_user_logged_in');
add_action('wp_ajax_nopriv_is_user_logged_in', 'ajax_check_user_logged_in');
?>

var data = {
    action: 'is_user_logged_in'
};

jQuery.post(ajaxurl, data, function(response) {
    if(response == 'yes') {
        // user is logged in, do your stuff here
    } else {
        // user is not logged in, show login form here
    }
});